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THE  HUMAN  WORTH  OF  RIGOROUS  THINKING 


8  tit  t)  or 


SCIENCE  AND  RELIGION:  THE  RATIONAL  AND  THE 
STTPERRATIONAL  The  Yale  University  Press 

THE  NEW  INFINITE  AND  THE  OLD  THEOLOGY 

The  Yale  University  Press 


COLUMBIA   UNIVERSITY  PRESS 

SALES  AGENTS 

NEW  YORK  LONDON 

LEMCKE  AND  BUECHNER  HUMPHREY  MILFORD 

30-32  WEST  27TH  STREET  AMEN  CORNER,  E.  C. 


THE  HUMAN  WORTH  OF 
RIGOROUS  THINKING 

ESSAYS  AND  ADDRESSES 


BY 


CASSIUS   J.   KEYSER,   PH.D.,   LL.D. 

ADRA1N   PROFESSOR  OF  MATHEMATICS 
COLUMBIA   UNIVERSITY 


flrto 
COLUMBIA  UNIVERSITY  PRESS 

1916 
Alt  rights  reserved 

PRINTED  IN  U.  •    ft? 


Copyright,  1916 
BY  COLUMBIA  UNIVERSITY  PRESS 


Printed  from  type,  May,  1916 


PREFACE 

THE  following  fifteen  essays  and  addresses  have  ap- 
peared, in  the  course  of  the  last  fifteen  years,  as  articles 
in  various  scientific,  literary,  and  philosophical  journals. 
For  permission  to  reprint  I  have  to  thank  the  editors  and 
managers  of  The  Columbia  University  Quarterly,  The  Co- 
lumbia University  Press,  Science,  The  Educational  Review, 
The  Bookman,  The  Monist,  The  Hibbert  Journal,  and  The 
Journal  of  Philosophy,  Psychology  and  Scientific  Methods. 

The  title  of  the  volume  indicates  its  subject.  The  fact 
that  one  of  the  essays,  the  initial  one,  bears  the  same  title 
is  hardly  more  than  a  mere  coincidence,  for  all  of  the  dis- 
cussions deal  with  the  subject  in  question  and  nearly  all 
of  them  deal  with  it  directly,  consciously,  and  in  terms. 

In  passing  from  essay  to  essay  the  attentive  reader  will 
notice  a  few  repetitions  of  thought  and  possibly  a  few 
in  forms  of  expression.  Such  reiterations,  which  owe 
their  presence  to  the  occasional  character  of  the  essays 
and  to  the  aims  and  circumstances  that  originally  con- 
trolled their  composition,  may,  it  is  hoped,  be  regarded 
by  the  charitable  reader  less  as  blemishes  than  as  means 
of  emphasizing  important  considerations. 

CASSIUS  J.  KEYSER. 
April  14,  1916. 


CONTENTS 

CHAPTER  PAGE 

I.  The  Human  Worth  of  Rigorous  Thinking i 

II.  The  Human  Significance  of  Mathematics 26 

m.  The  Humanization  of  the  Teaching  of  Mathematics 61 

IV.  The  Walls  of  the  World;   or  Concerning  the  Figure  and  the 

Dimensions  of  the  Universe  of  Space 81 

V.  Mathematical  Emancipations:  Dimensionality  and  Hyperspace  xox 

VI.  The  Universe  and  Beyond:  The  Existence  of  the  Hypercosmic  122 

VII.  The  Axiom  of  Infinity:  A  New  Presupposition  of  Thought    .  139 

VIII.  The  Permanent  Basis  of  a  Liberal  Education 163 

DC.  Graduate  Mathematical  Instruction  for  Graduate  Students  not 

Intending  to  Become  Mathematicians 176 

X.  The  Source  and  Functions  of  a  University 201 

XI.   Research  in  American  Universities 209 

XII.  Principia  Mathematica 220 

XIII.  Concerning  Multiple  Interpretations  of  Postulate  Systems  and 

the  "  Existence  "  of  Hyperspace 233 

XIV.  Mathematical  Productivity  in  the  United  States 257 

XV.  Mathematics 271 


THE  HUMAN  WORTH   OF  RIGOROUS 
THINKING1 

But  in  the  strong  recess  of  Harmony 
Established  firm  abides  the  rounded  Sphere. 

—  EMPEDOCLES 

NEXT  to  the  peaceful  pleasure  of  meeting  genuine 
curiosity,  half-way,  upon  its  own  ground,  comes  the 
joy  of  combat  when  an  attack  upon  some  valued  right 
or  precious  interest  of  the  human  spirit  requires  to  be 
repelled.  Indeed,  given  a  competent  jury,  hardly  any 
other  undertaking  could  be  more  stimulating  than  to 
defend  mathematics  from  a  charge  of  being  unworthy 
to  occupy,  in  the  hierarchy  of  arts  and  sciences,  the 
high  place  to  which,  from  the  earliest  times,  the  judg- 
ment of  mankind  has  assigned  it.  But,  unfortunately, 
no  such  accusation  has  been  brought,  brought,  that  is, 
by  persons  of  such  scientific  qualifications  as  to  give 
their  opinion  in  the  premises  weight  enough  to  call 
for  serious  consideration.  Mathematics  has  been  often 
praised  by  the  scientifically  incompetent;  it  has  not, 
so  far  as  I  am  aware,  been  dispraised,  or  its  worth 
challenged  or  denied,  by  the  scientifically  competent. 
The  age-long  immunity  of  mathematics  from  authorita- 

1  An  address  delivered  before  the  Mathematical  Colloquium  of  Columbia 
University,  October  13,  1913.  Printed,  with  slight  change,  in  Science, 
December  5,  1913;  also,  with  other  slight  changes,  printed  in  The  Columbia 
University  Quarterly,  June,  1914,  under  the  title  "The  Study  of  Mathe- 
matics." The  substance  of  the  address  was  delivered  before  the  mathe- 
matics section  of  the  California  High  School  Teachers  Association,  August, 
1915,  at  Berkeley,  California. 


2  THE   HUMAN   WORTH   OF   RIGOROUS   THINKING 

tive  arraignment,  and  the  high  estimation  in  which 
the  science  has  been  almost  universally  held  in  enlight- 
ened times  and  places,  unite  to  give  it  a  position  nearly, 
if  not  quite,  unique  in  the  history  of  criticism.  Perhaps 
it  were  better  not  so.  Mathematicians  have  a  sense 
of  security  to  which,  it  may  be,  they  are  not  entitled 
in  a  critical  age  and  a  reeling  world.  Conceivably  it 
might  have  been  to  the  advantage  of  mathematics  and 
not  only  of  mathematics  but  of  science  in  general,  of 
philosophy,  too,  and  the  general  enlightenment,  if  in 
course  of  the  centuries  mathematicians  had  been  now 
and  then  really  compelled  by  adverse  criticism  of  their 
science  to  discover  and  to  present  not  only  to  themselves 
but  acceptably  to  their  fellow-men  the  deeper  justifica- 
tions, if  such  there  be,  of  the  world's  approval  and 
applause  of  their  work.  However  that  may  be,  no 
one  is  likely  to  dissent  from  the  opinion  of  Mr.  Bertrand 
Russell  that  "in  regard  to  every  form  of  activity  it  is 
necessary  that  the  question  should  be  asked  from  time 
to  time:  what  is  its  purpose  and  ideal?  In  what  way 
does  it  contribute  to  the  beauty  of  human  existence?" 
An  inquiry  that  is  thus  necessary  for  the  general  wel- 
fare ought  to  be  felt  as.  a  duty,  unless,  more  fortunately, 
it  be  felt  as  a  pleasure. 

Why  study  mathematics?  What  are  the  rightful 
claims  of  the  science  to  human  regard?  What  are  the 
grounds  upon  which  a  university  may  justify  the  annual 
expenditure  of  thirty  to  fifty  thousands  of  dollars  to 
provide  for  mathematical  instruction  and  mathematical 
research? 

A  slight  transformation  of  the  questions  will  help  to 
.disclose  their  significance  and  may  give  a  quicker  sense 
of  their  poignancy  and  edge.  What  is  mathematics? 
I  hasten  to  say  that  I  do  not  intend  to  detain  the  reader 


THE  HUMAN   WORTH   OP   RIGOROUS  THINKING  3 

and  thus  perhaps  to  dampen  his  interest  with  a  defini- 
tion of  mathematics,  though  it  must  be  said  that  the 
discovery  of  what  mathematics  is,  is  doubtless  one  of  the 
very  great  scientific  achievements  of  the  nineteenth 
century.  The  question  asks,  not  for  a  definition  of  the 
science,  but  for  a  brief  and  helpful  description  of  it  — 
for  an  obvious  mark  or  aspect  of  it  that  will  enable  us 
to  know  what  it  is  that  we  are  here  writing  or  reading 
about.  Well,  mathematics  may  be  viewed  either  as  an 
enterprise  or  as  a  body  of  achievements.  As  an  enter- 
prise mathematics  is  characterized  by  its  aim,  and  its 
aim  is  to  think  rigorously  whatever  is  rigorously  think- 
able or  whatever  may  become  rigorously  thinkable  in 
course  of  the  upward  striving  and  refining  evolution  of 
ideas.  As  a  body  of  achievements  mathematics  con- 
sists of  all  the  results  that  have  come,  in  the  course 
of  the  centuries,  from  the  prosecution  of  that  enter- 
prise: the  truth  discovered  by  it;  the  doctrines  created 
by  it;  the  influence  of  these,  through  their  applications 
and  their  beauty,  upon  the  advancement  of  civiliza- 
tion and  the  weal  of  man. 

Our  questions  now  stand:  Why  should  a  human  being 
desire  to  share  in  that  spiritual  enterprise  which  has 
for  its  aim  to  think  rigorously  whatever  is  or  may 
become  rigorously  thinkable  and  to  "frame  a  world 
according  to  a  rule  of  divine  perfection"?  Why  should 
men  and  women  seek  some  knowledge  of  that  variety 
of  perfection  with  which  men  and  women  have  enriched 
life  and  the  world  by  rigorous  thought?  What  are  the 
just  claims  to  human  regard  of  perfect  thought  and  the 
spirit  of  perfect  thinking?  Upon  what  grounds  may  a 
university  justify  the  annual  expenditure  of  thirty  to 
fifty  thousands  of  dollars  to  provide  for  the  disciplining 
of  men  and  women  in  the  art  of  thinking  rigorously 


4  THE   HUMAN   WORTH   OF   RIGOROUS   THINKING 

and  for  the  promotion  of  research  in  the  realm  of  exact 
thought? 

Such  are  the  questions.  They  plainly  sum  themselves 
in  one:  among  the  human  agencies  that  ameliorate  life, 
what  is  the  rdle  of  rigorous  thinking?  What  is  the  role 
of  the  spirit  that  always  aspires  to  the  attainment  of 
logical  perfection? 

Evidently  that  question  is  not  one  for  adequate 
handling  in  a  brief  magazine  article  by  an  ordinary 
student  of  mathematics.  Rather  is  it  a  subject  for  a 
long  series  of  lectures  by  a  learned  professor  of  the 
history  of  civilization.  Indeed  so  vast  is  the  subject 
that  even  an  ordinary  student  of  mathematics  can 
detect  some  of  the  more  obvious  tasks  such  a  philosophic 
historian  would  have  to  perform  and  a  few  of  the  dif- 
ficulties he  would  doubtless  encounter.  It  may  be 
worth  while  to  mention  some  of  them. 

Certainly  one  of  the  tasks,  and  probably  one  of  the 
difficulties  also,  would  be  that  of  securing  an  audience 
—  an  audience,  I  mean,  capable  of  understanding  the 
lectures,  for  is  not  a  genuine  auditor  a  listener  who 
understands?  To  understand  the  lectures  it  would 
seem  to  be  necessary  to  know  what  that  is  which  the 
lectures  are  about  —  that  is,  it  would  be  necessary  to 
know  what  is  meant  by  rigorous  thinking.  To  know 
this,  however,  one  must  either  have  consciously  done 
some  rigorous  thinking  or  else,  at  the  very  least,  have 
examined  some  specimens  of  it  pretty  carefully,  just 
as,  in  order  to  know  what  good  art  is,  it  is,  in  general, 
essential  either  to  have  produced  good  art  or  to  have 
attentively  examined  some  specimens  of  it,  or  to  have 
done  both  of  these  things.  Here,  then,  at  the  outset 
our  historian  would  meet  a  serious  difficulty,  unless 
his  audience  chanced  to  be  one  of  mathematicians, 


THE   HUMAN   WORTH   OF   RIGOROUS   THINKING  5 

which  is  (unfortunately)  not  likely,  inasmuch  as  the 
great  majority  of  mathematicians  are  so  exclusively 
interested  in  mathematical  study  or  teaching  or  research 
as  to  be  but  little  concerned  with  the  philosophical 
question  of  the  human  worth  of  their  science.  It  is, 
therefore,  easy  to  see  how  our  lecturer  would  have  to 
begin. 

Ladies  and  gentlemen,  we  have  met,  he  would  say, 
we  have  met  to  open  a  course  of  lectures  dealing  with  the 
role  of  rigorous  thinking  in  the  history  of  civilization. 
In  order  that  the  course  may  be  profitable  to  you,  in 
order  that  it  may  be  a  course  in  ideas  and  not  merely 
or  mainly  a  verbal  course,  it  is  essential  that  you  should 
know  what  rigorous  thinking  is  and  what  it  is  not. 
Even  I,  your  speaker,  he  will  own,  might  reasonably 
be  held  to  the  obligation  of  knowing  that. 

It  is  reasonable,  ladies  and  gentlemen,  it  is  reasonable 
to  assume,  he  would  say,  that  in  the  course  of  your 
education  you  neglected  mathematics,  and  it  is  there- 
fore probable  or  indeed  quite  certain  that,  notwith- 
standing your  many  accomplishments,  you  do  not  quite 
know  or  rather,  perhaps  I  should  say,  you  are  very  far 
from  knowing  what  rigorous  thinking  is  or  what  it  is 
not.  Of  course,  as  you  know,  it  is,  generally  speaking, 
much  easier  to  tell  what  a  thing  is  not  than  to  tell  what 
it  is,  and  I  might,  he  would  say,  I  might  proceed  by 
way  of  a  preliminary  to  indicate  roughly  what  rigorous 
thinking  is  not.  Thus  I  might  explain  that  rigorous 
thinking,  though  much  of  it  has  been  done  in  the  world 
and  though  it  has  produced  a  large  literature,  is  never- 
theless a  relatively  rare  phenomenon.  I  might  point 
out  that  a  vast  majority  of  mankind,  a  vast  majority  of 
educated  men  and  women,  have  not  been  disciplined 
to  think  rigorously  even  those  things  that  are  most 


6  THE   HUMAN   WORTH   OF   RIGOROUS   THINKING 

available  for  such  thinking.  I  might  point  out  that, 
on  the  other  hand,  most  of  the  ideas  with  which  men 
and  women  have  constantly  to  deal  are  as  yet  too  nebu- 
lous and  vague,  too  little  advanced  in  the  course  of 
their  evolution,  to  be  available  for  concatenative  think- 
ing and  rigorous  discourse.  I  should  have  to  say,  he 
would  add,  that,  on  these  accounts,  most  of  the  think- 
ing done  in  the  world  in  a  given  day,  whether  done  by 
men  in  the  street  or  by  farmers  or  factory-hands  or 
administrators  or  historians  or  physicians  or  lawyers 
or  jurists  or  statesmen  or  philosophers  or  men  of  letters 
or  students  of  natural  science  or  even  mathematicians 
(when  not  strictly  employed  in  their  own  subject), 
comes  far  short  of  the  demands  and  standards  of  rig- 
orous thinking. 

I  might  go  on  to  caution  you,  our  speaker  would  say, 
against  the  current  fallacy,  recently  advanced  by  elo- 
quent writers  to  the  dignity  of  a  philosophical  tenet, 
of  regarding  what  is  called  successful  action  as  the 
touchstone  of  rigorous  thinking.  For  you  should  know 
that  much  of  what  passes  in  the  world  for  successful 
action  proceeds  from  impulse  or  instinct  and  not  from 
thinking  of  any  kind;  .you  should  know  that  no  action 
under  the  control  of  non-rigorous  thinking  can  be 
strictly  successful  except  by  the  favor  of  chance  or 
through  accidental  compensation  of  errors;  you  should 
know  that  most  of  what  passes  for  successful  action, 
most  of  what  the  world  applauds  and  even  commem- 
morates  as  successful  action,  so  far  from  being  really 
successful,  varies  from  partial  failure  to  failure  that, 
if  not  total,  would  at  all  events  be  fatal  in  any  universe 
that  had  the  economic  decency  to  forbid,  under  pain 
of  death,  the  unlimited  wasting  of  its  resources.  The 
dominant  animal  of  such  a  universe  would  be,  in  fact, 


THE   HUMAN   WORTH   OF   RIGOROUS   THINKING  ^ 

a  superman.  In  our  world  the  natural  resources  of 
life  are  superabundant,  and  man  is  poor  in  reason 
because  he  has  been  the  prodigal  son  of  a  too  opulent 
mother.  But,  ladies  and  gentlemen,  our  speaker  will 
conclude,  you  will  know  better  what  rigorous  thinking 
is  not  when  once  you  have  learned  what  it  is.  This, 
however,  cannot  well  be  learned  in  a  course  of  lectures 
in  which  that  knowledge  is  presumed.  I  have,  there- 
fore, to  adjourn  this  course  until  such  time  as  you  shall 
have  gained  that  knowledge.  It  cannot  be  gained  by 
reading  about  it  or  hearing  about  it.  The  easiest  way, 
for  most  persons  the  only  way,  to  gain  it  is  to  examine 
with  exceeding  patience  and  care  some  specimens,  at 
least  one  specimen,  of  the  literature  in  which  rigorous 
thinking  is  embodied.  Such  a  specimen,  he  could 
add,  is  Dr.  Thomas  L.  Heath's  magnificent  edition  of 
Euclid,  where  an  excellent  translation  of  the  Elements 
from  the  definitive  text  of  Heiberg  is  set  in  the  composite 
light  of  critical  commentary  from  Aristotle  down  to 
the  keenest  logical  microscopists  and  histologists  of  our 
own  day.  If  you  think  Euclid  too  ancient  or  too  stale 
even  when  seasoned  with  the  wit  of  more  than  two 
thousand  years  of  the  acutest  criticism,  you  may  find 
a  shorter  and  possibly  a  fresher  way  by  examining 
minutely  such  a  work  as  Veronese's  GrundzUge  der 
Geometric  or  Hilbert's  famous  Foundations  of  Geom- 
etry or  Peano's  Sui  Numeri  Irrazionali.  In  works  of 
this  kind  and  not  elsewhere  you  will  find  in  its  nakedness, 
purity,  and  spirit,  what  you  have  neglected  and  what 
you  need.  You  will  note  that  in  the  beginning  of  such 
a  work  there  is  found  a  system  of  assumptions  or  postu- 
lates, discovered  the  Lord  only  and  a  few  men  of  genius 
know  where  or  how,  selected  perhaps  with  reference  to 
simplicity  and  clearness,  certainly  selected  and  tested 


8  THE  HUMAN  WORTH   OF   RIGOROUS   THINKING 

with  respect  to  their  compatibility  and  independence, 
and,  it  may  be,  with  respect  also  to  categoricity.  You 
will  not  fail  to  observe  with  the  utmost  minuteness, 
and  from  every  possible  angle,  how  it  is  that  upon  these 
postulates  as  a  basis  there  is  built  up  by  a  kind  of 
divine  masonry,  little  step  by  step,  a  stately  struc- 
ture of  ideas,  an  imposing  edifice  of  rigorous  thought, 
a  towering  architecture  of  doctrine  that  is  at  once 
beautiful,  austere,  sublime,  and  eternal.  Ladies  and 
gentlemen,  our  speaker  will  say,  to  accomplish  that  ex- 
amination will  require  twelve  months  of  pretty  assiduous 
application.  The  next  lecture  of  this  course  will  be 
given  one  year  from  date. 

On  resuming  the  course  what  will  our  philosopher 
and  historian  proceed  to  say?  He  will  begin  to  say  what, 
if  he  says  it  concisely,  will  make  up  a  very  large  vol- 
ume. Room  is  lacking  here,  even  if  competence  were 
not,  for  so  much  as  an  adequate  outline  of  the  matter. 
It  is  possible,  however,  to  draw  with  confidence  a  few 
of  the  larger  lines  that  such  a  sketch  would  have  to 
contain. 

What  is  it  that  our  speaker  will  be  obliged  to  deal 
with  first?  I  do  not  mean  obliged  logically  nor  obliged 
by  an  orderly  development  of  his  subject.  I  mean 
obliged  by  the  expectation  of  his  hearers.  Every  one 
can  answer  that  question.  For  presumably  the  audience 
represents  the  spirit  of  the  times,  and  this  age  is,  at 
least  to  a  superficial  observer,  an  age  of  engineering. 
Now,  what  is  engineering?  Well,  the  Charter  of  the 
Institution  of  Civil  Engineers  tells  us  that  engineering 
is  the  "art  of  directing  the  great  sources  of  power  in 
Nature  for  the  use  and  convenience  of  man."  By 
Nature  here  must  be  meant  external  or  physical  nature, 
for,  if  internal  nature  were  also  meant,  every  good  form 


THE   HUMAN   WORTH  OF   RIGOROUS  THINKING  9 

of  activity  would  be  a  species  of  engineering,  and  maybe 
it  is  such,  but  that  is  a  claim  which  even  engineers 
would  hardly  make  and  poets  would  certainly  deny. 
Use  and  convenience — these  are  the  key-bearing  words. 
It  is  perfectly  evident  that  our  lecturer  will  have  to  deal 
first  of  all  with  what  the  world  would  call  the  "utility" 
of  rigorous  thinking,  that  is  to  say,  with  the  applica- 
tions of  mathematics  and  especially  with  its  applica- 
tions to  problems  of  engineering.  If  he  really  knows 
profoundly  what  mathematics  is,  he  will  not  wish  to 
begin  with  applications  nor  even  to  make  applications 
a  major  theme  of  his  discourse,  but  he  must,  and  he  will 
do  so  uncomplainingly  as  a  concession  to  the  external- 
mindedness  of  his  time  and  his  audience.  He  will  not  only 
desire  to  show  his  audience  applications  of  mathematics 
to  engineering,  but,  being  an  historian  of  civilization,  he 
will  especially  desire  to  show  them  the  development  of 
such  applications  from  the  earliest  times,  from  the 
building  of  pyramids  and  the  mensuration  of  land  in 
ancient  Egypt  down  to  such  splendid  modern  achieve- 
ments as  the  designing  and  construction  of  an  Eads 
Bridge,  an  ocean  Imperator  or  a  Panama  Canal.  The 
story  will  be  long  and  difficult,  but  it  will  edify.  The 
audience  will  be  amazed  at  the  truth  if  they  under- 
stand. If  they  do  not  understand  the  truth  fully,  our 
speaker  must  at  all  events  contrive  that  they  shall  see 
it  in  glimmers  and  gleams  and,  above  all,  that  they  shall 
acquire  a  feeling  for  it.  They  must  be  led  to  some 
acquaintance  with  the  great  engineering  works  of  the 
world,  past  and  present;  they  must  be  given  an  intel- 
ligible conception  of  the  immeasurable  contribution 
such  works  have  made  to  the  comfort,  convenience,  and 
power  of  man;  and  especially  must  they  be  convinced 
of  the  fact  that,  not  only  would  the  greatest  of  such 


10          THE   HUMAN   WORTH   OF   RIGOROUS   THINKING 

achievements  have  been,  except  for  mathematics,  utterly 
impossible,  but  that  such  of  the  lesser  ones  as  could 
have  been  wrought  without  mathematical  help  could 
not  have  been  thus  accomplished  without  wicked  and 
pathetic  waste  both  of  material  resources  and  of  human 
toil.  In  respect  to  this  latter  point,  the  relation  of 
mathematics  to  practical  economy  in  large  affairs,  our 
speaker  will  no  doubt  invite  his  hearers  to  read  and 
reflect  upon  the  ancient  work  of  Frontinus  on  the  Water 
Supply  of  the  City  of  Rome  in  order  that  thus  they  may 
gain  a  vivid  idea  of  the  fact  that  the  most  practical 
people  of  history,  despising  mathematics  and  the  finer 
intellectualizations  of  the  Greeks,  were  unable  to  accom- 
plish their  own  great  engineering  feats  except  through 
appalling  waste  of  materials  and  men.  Our  lecturer 
will  not  be  content,  however,  with  showing  the  service 
of  mathematics  in  the  prevention  of  waste;  he  will 
show  that  it  is  indispensable  to  the  productivity  and 
trade  of  the  modern  world.  Before  quitting  this  divi- 
sion of  his  subject  he  will  have  demonstrated  that,  if 
all  the  contributions  which  mathematics  has  made,  and 
which  nothing  else  could  make,  to  navigation,  to  the 
building  of  railways,  to  the  construction  of  ships,  to  the 
subjugation  of  wind  and  wave,  electricity  and  heat,  and 
many  other  forms  and  manifestations  of  energy,  he 
will  have  demonstrated,  I  say,  and  the  audience  will 
finally  understand,  that,  if  all  these  contributions  of 
mathematics  were  suddenly  withdrawn,  the  life  and  body 
of  industry  and  commerce  would  suddenly  collapse  as 
by  a  paralytic  stroke,  the  now  splendid  outer  tokens 
of  material  civilization  would  perish,  and  the  face  of 
our  planet  would  quickly  assume  the  aspect  of  a  ruined 
and  bankrupt  world. 

As  our  lecturer  has  been  constrained  by  circumstances 


HUMAN   WORTH   OF   RIGOROUS  THINKING  II 

to  back  into  his  subject,  as  he  has,  that  is,  been  com- 
pelled to  treat  first  of  the  service  that  mathematics  has 
rendered  engineering,  he  will  probably  next  speak  of 
the  applications  of  mathematics  to  the  so-called  natural 
sciences  —  the  more  properly  called  experimental  sciences 
—  of  physics,  chemistry,  biology,  economics,  psychology, 
and  the  like.  Here  his  task,  if  it  is  to  be,  as  it  ought  to 
be,  expository  as  well  as  narrative,  will  be  exceedingly 
hard.  For  how  can  he  weave  into  his  narrative  an  intel- 
ligible exposition  of  Newton's  Principia,  Laplace's  Me- 
canique  Celeste,  Lagrange's  Mecanique  Analytique,  Gauss's 
Theoria  Motus  Corporum  Coelestium,  Fourier's  Thtorie 
Analytique  de  la  Chaleur,  Maxwell's  Electricity  and 
Magnetism,  not  to  mention  scores  of  other  equally  dif- 
ficult and  hardly  less  important  works  of  a  mathemat- 
ical-physical character?  Even  if  our  speaker  knew  it 
all,  which  no  man  can,  he  could  not  tell  it  all  in- 
telligibly to  his  hearers.  These  will  have  to  be  con- 
tent with  a  rather  general  and  superficial  view,  with  a 
somewhat  vague  intuition  of  the  truth,  with  fragmentary 
and  analogical  insights  gained  through  settings  forth  of 
great  things  by  small;  and  they  will  have  to  help  them- 
selves and  their  speaker,  too,  by  much  pertinent  read- 
ing. No  doubt  the  speaker  will  require  his  hearers, 
in  order  that  they  may  thus  gain  a  tolerable  perspective, 
to  read  well  not  only  the  first  two  volumes  of  the 
magnificent  work  of  John  Theodore  Merz  dealing  with 
the  History  of  European  Thought  in  the  Nineteenth  Cen- 
tury, but  also  many  selected  portions  of  the  kindred 
literature  there  cited  in  richest  profusion.  The  work 
treats  mainly  of  natural  science,  but  it  deals  with  it 
philosophically,  under  the  larger  aspect,  that  is,  of 
science  regarded  as  Thought.  By  the  help  of  such 
literature  in  the  hands  of  his  auditors,  our  lecturer  will 


12          THE   HUMAN   WORTH   OF   RIGOROUS   THINKING 

be  able  to  give  them  a  pretty  vivid  sense  of  the  great 
and  increasing  role  of  mathematics  in  suggesting,  formu- 
lating, and  solving  problems  in  all  branches  of  natural 
science.  Whether  it  be  with  "the  astronomical  view 
of  nature"  that  he  is  dealing,  or  "the  atomic  view"  or 
"the  mechanical  view"  or  "the  physical  view"  or 
"the  morphological  view"  or  "the  genetic  view"  or  "the 
vitalistic  view"  or  "the  psychophysical  view"  or  "the 
statistical  view,"  in  every  case,  in  all  these  great  at- 
tempts of  reason  to  create  or  to  find  a  cosmos  amid  the 
chaos  of  the  external  world,  the  presence  of  mathe- 
matics and  its  manifold  service,  both  as  instrument  and 
as  norm,  illustrate  and  confirm  the  Kantian  and  Rie- 
mannian  conception  of  natural  science  as  "the  attempt 
to  understand  nature  by  means  of  exact  concepts." 

In  connection  with  this  division  of  his  subject,  our 
speaker  will  find  it  easy  to  enter  more  deeply  into  the 
spirit  and  marrow  of  it.  He  will  be  able  to  make  it 
clear  that  there  is  a  sense,  a  just  and  important  sense, 
in  which  all  thinkers  and  especially  students  of  natural 
science,  though  their  thinking  is  for  the  most  part  not 
rigorous,  are  yet  themselves  contributors  to  mathematics. 
I  do  not  refer  to  the  powerful  stimulation  of  mathe- 
matics by  natural  science  in  furnishing  it  with  many  of 
its  problems  and  in  constantly  seeking  its  aid.  What 
I  mean  is  that  all  thinkers  and  especially  students  of 
natural  science  are  engaged,  both  consciously  and  un- 
consciously, both  intentionally  and  unintentionally, 
in  the  mathematicization  of  concepts  —  that  is  to  say, 
in  so  transforming  and  refining  concepts  as  to  fit  them 
finally  for  the  amenities  of  logic  and  the  austerities  of 
rigorous  thinking,  We  are  dealing  here,  our  speaker 
will  say,  with  a  process  transcending  conscious  design. 
We  are  dealing  with  a  process  deep  in  the  nature  and 


THE  HUMAN   WORTH   OF   RIGOROUS  THINKING          13 

being  of  the  psychic  world.  Like  a  child,  an  idea,  once 
it  is  born,  once  it  has  come  into  the  realm  of  spiritual 
light,  possibly  long  before  such  birth,  enters  upon  a 
career,  a  career,  however,  that,  unlike  the  child's,  seems 
to  be  immortal.  In  most  cases  and  probably  in  all, 
an  idea,  on  entering  the  world  of  consciousness,  is  vague, 
nebulous,  formless,  not  at  once  betraying  either  what 
it  is  or  what  it  is  destined  to  become.  Ideas,  however, 
are  under  an  impulse  and  law  of  amelioration.  The 
path  of  their  upward  striving  and  evolution  —  often 
a  long  and  winding  way  —  leads  towards  precision  and 
perfection  of  form.  The  goal  is  mathematics.  Witness, 
for  example,  our  lecturer  will  say,  the  age-long  travail 
and  aspiration  of  the  great  concept  now  known  as  mathe- 
matical continuity,  a  concept  whose  inner  structure  is 
even  now  known  and  understood  only  of  mathematicians, 
though  the  ancient  Greeks  helped  in  molding  its  form 
and  though  it  has  long  been,  if  somewhat  blindly,  yet 
constantly  employed  in  natural  science,  as  when  a 
physicist,  for  example,  or  an  astronomer  uses  such 
numbers  as  e  and  tr  in  computation.  Witness,  again, 
how  that  supreme  concept  of  mathematics,  the  concept 
of  function,  has  struggled  through  thousands  of  years 
to  win  at  length  its  present  precision  of  form  out  of 
the  nebulous  sense,  which  all  minds  have,  of  the  mere 
dependence  of  things  on  other  things.  Witness,  too,  he 
will  say,  the  mathematical  concept  of  infinity,  which 
prior  to  a  half-century  ago  was  still  too  vague  for  logical 
discourse,  though  from  remotest  antiquity  the  great 
idea  has  played  a  conspicuous  role,  mainly  emotional, 
in  theology,  philosophy,  and  science.  Like  examples 
abound,  showing  that  one  of  the  most  impressive  and 
significant  phenomena  in  the  life  of  the  psychic  world, 
if  we  will  but  discern  and  contemplate  it,  is  the  process 


14          THE   HUMAN   WORTH   OF   RIGOROUS   THINKING 

by  which  ideas  advance,  often  slowly  indeed  but  surely, 
from  their  initial  condition  of  formlessness  and  inde- 
termination  to  the  mathematical  estate.  The  chemic- 
ization  of  biology,  the  physicization  of  chemistry,  the 
mechanicization  of  physics,  the  mathematicization  of 
mechanics,  the  arithmeticization  of  mathematics,  these 
well-known  tendencies  and  drifts  in  science  do  but  illus- 
trate on  a  large  scale  the  ubiquitous  process  in  question. 

At  length,  ladies  and  gentlemen,  our  speaker  will  say, 
in  the  light  of  the  last  consideration  the  deeper  and 
larger  aspects  of  our  subject  are  beginning  to  show 
themselves  and  there  is  dawning  upon  us  an  impressive 
vision.  The  nature,  function,  and  life  of  the  entire 
conceptual  world  seem  to  come  within  the  circle  and 
scope  of  our  present  enterprise.  We  are  beginning  to 
see  that  to  challenge  the  human  worth  of  mathematics, 
to  challenge  the  worth  of  rigorous  thinking,  is  to  chal- 
lenge the  worth  of  all  thinking,  for  now  we  see  that 
mathematics  is  but  the  ideal  to  which  all  thinking,  by 
an  inevitable  process  and  law  of  the  human  spirit, 
constantly  aspires.  We  see  that  to  challenge  the  worth 
of  that  ideal  is  to  arraign  before  the  bar  of  values  what 
seems  the  deepest  process  and  inmost  law  of  the  uni- 
verse of  thought.  Indeed  we  see  that  in  defending 
mathematics  we  are  really  defending  a  cause  yet  more 
momentous,  the  whole  cause,  namely,  of  the  conceptual 
procedure  of  science  and  the  conceptual  activity  of  the 
human  mind,  for  mathematics  is  nothing  but  such  con- 
ceptual procedure  and  activity  come  to  its  maturity, 
purity,  and  perfection. 

Now,  ladies  and  gentlemen,  our  lecturer  will  say,  I 
cannot  in  this  course  deal  explicitly  and  fully  with  this 
larger  issue.  But,  he  will  say,  we  are  living  in  a  day 
when  that  issue  has  been  raised;  we  happen  to  be  living 


THE   HUMAN   WORTH   OP  RIGOROUS   THINKING  15 

in  a  time  when,  under  the  brilliant  and  effective  leader- 
ship of  such  thinkers  as  Professor  Bergson  and  the  late 
Professor  James,  the  method  of  concepts,  the  method  of 
intellect,  the  method  of  science,  is  being  powerfully 
assailed;  and,  he  will  say,  whilst  I  heartily  welcome  this 
attack  of  criticism  as  causing  scientific  men  to  reflect 
more  deeply  upon  the  method  of  science,  as  exhibiting 
more  clearly  the  inherent  limitations  of  its  method,  and 
as  showing  that  life  is  so  rich  as  to  have  many  precious 
interests  and  the  world  much  truth  beyond  the  reach 
of  that  method,  yet  I  cannot  refrain,  he  will  say,  from 
attempting  to  point  out  what  seems  to  me  a  radical 
error  of  the  critics,  a  fundamental  error  of  theirs,  in 
respect  to  what  is  the  highest  function  of  conception 
and  in  respect  to  what  is  the  real  aim  and  ideal  of  the 
life  of  intellect.  For  we  shall  thus  be  led  to  a  deeper 
view  of  our  subject  proper. 

These  critics  find,  as  all  of  us  find,  that  what  we  call 
mind  or  our  minds  is,  in  some  mysterious  way,  func- 
tionally connected  with  certain  living  organisms  known 
as  human  bodies;  they  find  that  these  living  bodies 
are  constantly  immersed  in  a  universe  of  matter  and 
motion  in  which  they  are  continually  pushed  and  pulled, 
heated  and  cooled,  buffeted  and  jostled  about  —  a 
universe  that,  according  to  James,  would,  in  the  "ab- 
sence of  concepts,"  reveal  itself  as  "a  big  blooming 
buzzing  confusion"  —  though  it  is  hard  to  see  how  such 
a  revelation  could  happen  to  any  one  devoid  of  the 
concept  "confusion,"  but  let  that  pass;  our  critics  find 
that  our  minds  get  into  some  initial  sort  of  knowing 
connection  with  that  external  blooming  confusion  through 
what  they  call  the  sensibility  of  our  bodies,  yielding 
all  manner  of  sensations  as  of  weights,  pressures,  pushes 
and  pulls,  of  intensities  and  extensities  of  brightness, 


1 6          THE,  HUMAN  WORTH   OF  RIGOROUS   THINKING 

sound,  time,  colors,  space,  odors,  tastes,  and  so  on;  they 
find  that  we  must,  on  pain  of  organic  extinction,  take 
some  account  of  these  elements  of  the  material  world; 
they  find  that,  as  a  fact,  we  human  beings  constantly 
deal  with  these  elements  through  the  instrumentality 
of  concepts;  they  find  that  the  effectiveness  of  our 
dealing  with  the  material  world  is  precisely  due  to  our 
dealing  with  it  conceptually;  they  infer  that,  there- 
fore, dealing  with  matter  is  exactly  what  concepts  are 
for,  saying  with  Ostwald,  for  example,  that  the  goal  of 
natural  science,  the  goal  of  the  conceptual  method  of 
mind,  "is  the  domination  of  nature  by  man";  not  only, 
our  speaker  will  say,  do  our  critics  find  that  we  deal 
with  the  material  world  conceptually,  and  effectively 
because  conceptually,  but  they  find  also  that  life  has 
interests  and  the  world  values  not  accessible  to  the  con- 
ceptual method,  and  as  this  method  is  the  method  of 
the  intellect,  they  conclude,  not  only  that  the  intellect 
cannot  grasp  life,  but  that  the  aim  and  ideal  of  intellect 
is  the  understanding  and  subjugation  of  matter,  saying 
with  Professor  Bergson  "that  our  intellect  is  intended 
to  think  matter,"  "that  our  concepts  have  been  formed 
on  the  model  of  solids,"  "that  the  essential  function 
of  our  intellect  .  .  .  is  to  be  a  light  for  our  conduct, 
to  make  ready  for  our  action  on  things,"  that  "the 
intellect  always  behaves  as  if  it  were  fascinated  by  the 
contemplation  of  inert  matter,"  that  "intelligence  .  .  . 
aims  at  a  practically  useful  end,"  that  "the  intellect  is 
never  quite  at  its  ease,  .  .  .  except  when  it  is  working 
upon  inert  matter,  more  particularly  upon  solids,"  and 
much  more  to  the  same  effect. 

Now,  ladies  and  gentlemen,  our  speaker  will  ask, 
what  are  we  to  think  of  this?  What  are  we  to  think  of 
this  evaluation  of  the  science-making  method  of  con- 


THE   HUMAN   WORTH   OF   RIGOROUS   THINKING  17 

cepts?  What  are  we  to  think  of  the  aim  and  ideal 
here  ascribed  to  the  intellect  and  of  the  station  assigned 
it  among  the  faculties  of  the  human  mind?  In  the  first 
place,  he  will  say,  it  ought  to  be  evident  to  the  critics 
themselves,  and  evident  to  them  even  in  what  they 
esteem  the  poor  light  of  intellect,  that  the  above- 
sketched  movement  of  their  minds  is  a  logically  unsound 
movement.  They  do  not  indeed  contend  that,  because 
a  living  being  in  order  to  live  must  deal  with  the  material 
world,  it  must,  therefore,  do  so  by  means  of  concepts. 
The  lower  animals  have  taught  them  better.  But 
neither  does  it  follow  that,  because  certain  bipeds  in 
dealing  with  the  material  world  deal  with  it  concep- 
tually, the  essential  function  of  concepts  is  just  to  deal 
with  matter.  Nor  does  such  an  inference  respecting 
the  essential  function  of  concepts  follow  from  the  fact 
that  the  superior  effectiveness  of  man's  dealing  with  the 
physical  world  is  due  to  his  dealing  with  it  conceptually. 
For  it  is  obviously  conceivable  and  supposable  that 
such  conceptual  dealing  with  matter  is  only  an  incident 
or  byplay  or  subordinate  interest  in  the  career  of  con- 
cepts. It  is  conceivably  possible  that  such  employ- 
ment with  matter  is  only  an  avocation,  more  or  less 
serious  indeed  and  more  or  less  advantageous,  yet  an 
avocation,  and  not  the  vocation,  of  intellect.  Is  it 
not  evidently  possible  to  go  even  further?  Is  it  not 
logically  possible  to  admit  or  to  contend  that,  inasmuch 
as  the  human  intellect  is  functionally  attached  to  a 
living  body  which  is  itself  plunged  in  a  physical  uni- 
verse, it  is  absolutely  necessary  for  the  intellect  to  con- 
cern itself  with  matter  in  order  to  preserve,  not  indeed 
the  animal  life  of  man,  but  his  intellectual  life  —  is  it 
not  allowable,  he  will  say,  to  admit  or  to  maintain  that 
and  at  the  same  time  to  deny  that  such  concernment 


1 8          THE   HUMAN  WORTH   OF   RIGOROUS   THINKING 

with  matter  is  the  intellect's  chief  or  essential  function 
and  that  the  subjugation  of  matter  is  its  ideal  and  aim? 

Of  course,  our  lecturer  will  say,  our  critics  might  be 
wrong  in  their  logic  and  right  in  their  opinion,  just  as 
they  might  be  wrong  in  their  opinion  and  right  in  their 
logic,  for  opinion  is  often  a  matter,  not  of  logic  or  proof, 
but  of  temperament,  taste,  and  insight.  But,  he  will 
say,  if  the  issue  as  to  the  chief  function  of  concepts  and 
the  ideal  of  the  intellect  is  to  be  decided  in  accordance 
with  temperament,  taste,  and  insight,  then  there  is  room 
for  exercise  of  the  preferential  faculty,  and  alternatives 
far  superior  to  the  choice  of  our  critics  are  easy  enough 
to  find.  It  may  accord  better  with  our  insight  and 
taste  to  agree  with  Aristotle  that  "It  is  owing,"  not  to 
the  necessity  of  maintaining  animal  life  or  the  desire 
of  subjugating  matter,  but  "it  is  owing  to  their  wonder 
that  men  both  now  begin  and  first  began  to  philoso- 
phize; they  wondered  originally  at  the  obvious  diffi- 
culties, then  advanced  little  by  little  and  stated  the 
difficulties  about  the  greater  matters."  The  striking 
contrast  of  this  with  the  deliverances  of  Bergson  is 
not  surprising,  for  Aristotle  was  a  pupil  of  Plato  and 
the  doctrine  of  Bergson  is  that  of  Plato  completely 
inverted.  It  may  accord  better  with  our  insight  and 
taste  to  agree  with  the  great  K.  G.  J.  Jacobi,  who,  when 
he  had  been  reproached  by  Fourier  for  not  devoting  his 
splendid  genius  to  physical  investigations  instead  of 
pure  mathematics,  replied  that  a  philosopher  like  his 
critic  "ought  to  know  that  the  unique  end  of  science 
is,"  not  public  utility  and  application  to  natural  phe- 
nomena, but  "is  the  honor  of  the  human  spirit."  It 
may  accord  better  with  our  temperament  and  insight 
to  agree  with  the  sentiment  of  Diotima:  "I  am  per- 
suaded that  all  men  do  all  things,  and  the  better  they 


THE  HUMAN   WORTH   OF   RIGOROUS  THINKING  19 

are,  the  better  they  do  them,  in  the  hope,"  not  of 
subjugating  matter,  but  "in  the  hope  of  the  glorious 
fame  of  immortal  virtue." 

But  it  is  unnecessary,  ladies  and  gentlemen,  it  is  un- 
necessary, our  speaker  will  say,  to  bring  the  issue  to 
final  trial  in  the  court  of  temperaments  and  tastes. 
We  should  gain  there  a  too  easy  victory.  The  critics 
are  psychologists,  some  of  them  eminent  psychologists. 
Let  the  issue  be  tried  in  the  court  of  psychology,  for 
it  is  there  that  of  right  it  belongs.  They  know  the 
fundamental  and  relevant  facts.  What  is  the  verdict 
according  to  these?  The  critics  know  the  experiments 
that  have  led  to  and  confirmed  the  psychophysical  law 
of  Weber  and  Fechner  and  the  doctrine  of  thresholds; 
they  know  that,  in  accordance  with  that  doctrine  and 
that  law,  an  appropriate  stimulus,  no  matter  what  the 
department  of  sense,  may  be  finite  in  amount  and  yet 
too  small,  or  finite  and  yet  too  large,  to  yield  a  sensa- 
tion; they  know  that  the  difference  between  two  stimuli 
of  a  kind  appropriate  to  a  given  sense  department,  no 
matter  what  department,  may  be  a  finite  difference  and 
yet  too  small  for  sensibility  to  detect,  or  to  work  a 
change  of  sensation;  they  ought  to  know,  though  they 
seem  not  to  have  recognized,  much  less  to  have  weighed, 
the  fact  that,  owing  to  the  presence  of  thresholds,  the 
greatest  number  of  distinct  sensations  possible  in  any 
department  of  sense  is  a  finite  number;  they  ought  to 
know  that  the  number  of  different  departments  of  sense 
is  also  a.  finite  number;  they  ought  to  know  that,  there- 
fore, the  total  number  of  distinct  or  different  sensations 
of  which  a  human  being  is  capable  is  a  finite  number; 
they  ought  to  know,  though  they  seem  not  to  have 
recognized  the  fact,  that,  on  the  other  hand,  the  world 
of  concepts  is  of  infinite  multiplicity,  that  concepts,  the 


20          THE   HUMAN   WORTH   OF   RIGOROUS   THINKING 

fruit  of  intellect,  as  distinguished  from  sensations,  the 
fruit  of  sensibility,  are  infinite  in  number;  they  ought, 
therefore,  to  see,  our  speaker  will  say,  though  none  of 
them  has  seen,  that  in  attempting  to  derive  intellect 
out  of  sensibility,  in  attempting  to  show  that  (as  James 
says)  "concepts  flow  out  of  percepts,"  they  are  con- 
fronted with  the  problem  of  bridging  the  immeasurable 
gulf  between  the  finite  and  the  infinite,  of  showing,  that 
is,  how  an  infinite  multiplicity  can  arise  from  one  that 
is  finite.  But  even  if  they  solved  that  apparently 
insuperable  problem,  they  could  not  yet  be  in  position 
to  affirm  that  the  function  of  intellect  and  its  concepts 
is,  like  that  of  sensibility,  just  the  function  of  dealing 
with  matter,  as  the  function  of  teeth  is  biting  and 
chewing.  Far  from  it. 

Let  us  have  another  look,  the  lecturer  will  say,  at 
the  psychological  facts  of  the  case.  Owing  to  the  pres- 
ence of  thresholds  in  every  department  of  sense  it  may 
happen  and  indeed  it  does  happen  constantly  in  every 
department,  that  three  different  amounts  of  stimulus  of 
a  same  kind  give  three  sensations  such  that  two  of  them 
are  each  indistinguishable  from  the  third  and  yet  are  dis- 
tinguishable from  one  -another.  Now,  for  sensibility  in 
any  department  of  sense,  two  magnitudes  of  stimulus 
are  unequal  or  equal  according  as  the  sensations  given 
by  them  are  or  are  not  distinguishable.  Accordingly  in 
the  world  of  sensible  magnitudes,  in  the  sensible  universe, 
in  the  world,  that  is,  of  felt  weights  and  thrusts  and 
pulls  and  pressures,  of  felt  brightnesses  and  warmths  and 
lengths  and  breadths  and  thicknesses  and  so  on,  in  this 
world,  which  is  the  world  of  matter,  magnitudes  are  such 
that  two  of  them  may  each  be  equal  to  a  third  without  being 
equal  to  one  another.  That,  our  speaker  will  say,  is  a 
most  significant  fact  and  it  means  that  the  sensible 


THE   HUMAN   WORTH   OF   RIGOROUS   THINKING          21 

world,  the  world  of  matter,  is  irrational,  infected  with 
contradiction,  contravening  the  essential  laws  of  thought. 
No  wonder,  he  will  say,  that  old  Heracleitus  declared 
the  unaided  senses  "give  a  fraud  and  a  lie." 

Now,  our  speaker  will  ask,  what  has  been  and  is  the 
behavior  of  intellect  in  the  presence  of  such  contra- 
diction? Observe,  he  will  say,  that  it  is  intellect,  and 
not  sensibility,  that  detects  the  contradiction.  Of  the 
irrationality  in  question  sensibility  remains  insensible. 
The  data  among  which  the  contradiction  subsists  are 
indeed  rooted  in  the  sensible  world,  they  inhere  in  the 
world  of  matter,  but  the  contradiction  itself  is  known 
only  to  the  logical  faculty  called  intellect.  Observe 
also,  he  will  say,  and  the  observation  is  important,  that 
such  contradictions  do  not  compel  the  intellect  to  any 
activity  whatever  intended  to  preserve  the  life  of  the 
living  organism  to  which  the  intellect  is  functionally 
attached.  That  is  a  lesson  we  have  from  our  physical 
kin,  the  beasts.  What,  then,  has  the  intellect  done 
because  of  or  about  the  contradiction?  Has  it  gone  on 
all  these  centuries,  as  our  critics  would  have  us  believe, 
trying  to  "think  matter,"  as  if  it  did  not  know  that 
matter,  being  irrational,  is  not  thinkable?  Far  from  it, 
he  will  say,  the  intellect  is  no  such  ass. 

What  it  has  done,  instead  of  endlessly  and  stupidly 
besieging  the  illogical  world  of  sensible  magnitudes 
with  the  machinery  of  logic,  what  it  has  done,  our  lec- 
turer will  say,  is  this:  it  has  created  for  itself  another 
world.  It  has  not  rationalized  the  world  of  sensible 
magnitudes.  That,  it  knows,  cannot  be  done.  It  has 
discerned  the  ineradicable  contradictions  inherent  in 
them,  and  by  means  of  its  creative  power  of  conception 
it  has  made  a  new  world,  a  world  of  conceptual  magni- 
tudes that,  like  the  continue  of  mathematics,  are  so 


22          THE   HUMAN   WORTH   OF   RIGOROUS   THINKING 

constructed  by  the  spiritual  architect  and  so  endowed 
by  it  as  to  be  free  alike  from  the  contradictions  of  the 
sensible  world  and  from  all  thresholds  that  could  give 
them  birth.  Indeed  conception,  to  speak  metaphorically 
hi  terms  borrowed  from  the  realm  of  sense,  is  a  kind  of 
infinite  sensibility,  transcending  any  finite  distinction, 
difference  or  threshold,  however  minute  or  fine.  And 
now,  our  speaker  will  say,  it  is  such  magnitudes,  magni- 
tudes created  by  intellect  and  not  those  discovered  by 
sense,  though  the  two  varieties  are  frequently  not 
discriminated  by  their  names;  it  is  such  conceptual  mag- 
nitudes that  constitute  the  subject-matter  of  science. 
If  the  magnitudes  of  science,  apart  from  their  ration- 
ality, often  bear  in  conformation  a  kind  of  close  resem- 
blance to  magnitudes  of  sense,  what  is  the  meaning  of 
the  fact?  It  means,  contrary  to  the  view  of  Bergson 
but  in  accord  with  that  of  Poincare,  that  the  free  crea- 
ative  artist,  intellect,  though  it  is  not  constrained,  yet 
has  chosen  to  be  guided,  in  so  far  as  its  task  allows, 
by  facts  of  sense.  Thus  we  have,  for  example,  concep- 
tual space  and  sensible  space  so  much  alike  in  conforma- 
tion that,  though  one  of  them  is  rational  and  the  other 
is  not,  the  undiscriminating  hold  them  as  the  same. 

And  now,  our  lecturer  will  ask,  for  we  are  nearing  the 
goal,  what,  then,  is  the  motive  and  aim  of  this  creative 
activity  of  the  intellect?  Evidently  it  is  not  to  preserve 
and  promote  the  life  of  the  human  body,  for  animals 
flourish  without  the  aid  of  concepts,  without  "discourse 
of  reason,"  and  despite  the  contradictions  in  the  world 
of  sense.  The  aim  is,  he  will  say,  to  preserve  and  pro- 
mote the  life  of  the  intellect  itself.  In  a  realm  infected 
with  irrationality,  with  omnipresent  contradictions  of 
the  laws  of  thought,  intellect  cannot  live,  much  less 
flourish;  in  the  world  of  sense,  it  has  no  proper  subject- 


THE  HUMAN   WORTH  OF   RIGOROUS   THINKING          23 

matter,  no  home,  no  life.  To  live,  to  flourish,  it  must 
be  able  to  think,  to  think  in  accordance  with  the  laws 
of  its  being.  It  is  stimulated  and  its  activity  is  sus- 
tained by  two  opposite  forces:  discord  and  concord. 
By  the  one  it  is  driven;  by  the  other,  drawn.  Intel- 
lect is  a  perpetual  suitor.  The  object  of  the  suit  is, 
not  the  conquest  of  matter,  it  is  a  thing  of  mind,  it  is 
the  music  of  the  spirit,  it  is  Harmonia,  the  beautiful 
daughter  of  the  Muses.  The  aim,  the  ideal,  the  beati- 
tude of  intellect  is  harmony.  That  is  the  meaning 
of  its  endless  talk  about  compatibilities,  consistencies 
and  concords,  and  that  is  the  meaning  of  its  endless 
battling  and  circumvention  and  transcendence  of  con- 
tradiction. But  what  of  the  applications  of  science  and 
public  service?  These  are  by-products  of  the  intellect's 
aim  and  of  the  pursuit  of  its  ideal.  Many  things  it 
regards  as  worthy,  high,  and  holy  —  applications  of 
science,  public  service,  the  "wonder"  of  Aristotle, 
Jacobi's  "honor  of  the  human  spirit,"  Diotima's  "glori- 
ous fame  of  immortal  virtue"  -but  that  which,  by 
the  law  of  its  being,  Intellect  seeks  above  all  and  per- 
petually pursues  and  loves,  is  Harmony.  It  is  for  a 
home  and  a  dwelling  with  her  that  intellect  creates  a 
world;  and  its  admonition  is:  Seek  ye  first  the  kingdom 
of  harmony,  and  all  these  things  shall  be  added  unto 
you. 

And  the  ideal  and  admonition,  thus  revealed  in  the 
light  of  analysis,  are.  justified  of  history.  Inverting  the 
order  of  time,  we  have  only  to  contemplate  the  great 
periods  in  the  intellectual  life  of  Paris,  Florence,  and 
Athens.  If,  among  these  mightiest  contributors  to  the 
spiritual  wealth  of  man,  Athens  is  supreme,  she  is  also 
supreme  in  her  devotion  to  the  intellect's  ideal.  It  is 
of  Athens  that  Euripides  sings: 


24  THE   HUMAN  WORTH   OF   RIGOROUS   THINKING 

The  sons  of  Erechtheus,  the  olden, 

Whom  high  gods  planted  of  yore 
In  an  old  land  of  heaven  upholden, 

A  proud  land  untrodden  of  war: 
They  are  hungered,  and  lo,  their  desire 

With  wisdom  is  fed  as  with  meat: 
In  their  skies  is  a  shining  of  fire, 

A  joy  in  the  fall  of  their  feet: 
And  thither  with  manifold  dowers, 

From  the  North,  from  the  hills,  from  the  morn, 
The  Muses  did  gather  their  powers, 

That  a  child  of  the  Nine  should  be  born; 
And  Harmony,  sown  as  the  flowers, 

Grew  gold  in  the  acres  of  corn. 

And  thus,  ladies  and  gentlemen,  our  lecturer  will  say, 
what  I  wish  you  to  see  here  is,  that  science  and  espe- 
cially mathematics,  the  ideal  form  of  science,  are  crea- 
tions of  the  intellect  in  its  quest  of  harmony.  It  is 
as  such  creations  that  they  are  to  be  judged  and  their 
human  worth  appraised.  Of  the  applications  of  mathe- 
matics to  engineering  and  its  service  in  natural  science, 
I  have  spoken  at  length,  he  will  say,  in  course  of  previous 
lectures.  Other  great  themes  of  our  subject  remain  for 
consideration.  To  appraise  the  worth  of  mathematics 
as  a  discipline  in  the  art  of  rigorous  thinking  and  as  a 
means  of  giving  facility  and  wing  to  the  subtler  imagina- 
tion; to  estimate  and  explain  its  value  as  a  norm  for 
criticism  and  for  the  guidance  of  speculation  and  pioneer- 
ing in  fields  not  yet  brought  under  the  dominion  of 
logic;  to  estimate  its  esthetic  worth  as  showing  forth 
in  psychic  light  the  law  and  order  of  the  psychic  world; 
to  evaluate  its  ethical  significance  in  rebuking  by  its 
certitude  and  eternality  the  facile  scepticism  that  doubts 
all  knowledge,  and  especially  in  serving  as  a  retreat  for 
the  spirit  when  as  at  times  the  world  of  sense  seems 
madly  bent  on  heaping  strange  misfortunes  up  and  "to 
and  fro  the  chances  of  the  years  dance  like  an  idiot  in 


THE   HUMAN   WORTH   OF   RIGOROUS    THINKING  25 

the  wind";  to  give  a  sense  of  its  religious  value  in  "the 
contemplation  of  ideas  under  the  form  of  eternity," 
in  disclosing  a  cosmos  of  perfect  beauty  and  everlasting 
order  and  in  presenting  there,  for  meditation,  endless 
sequences  traversing  the  rational  world  and  seeming  to 
point  to  a  mystical  region  above  and  beyond:  these  and 
similar  themes,  our  speaker  will  say,  remain  to  be  dealt 
with  in  subsequent  lectures  of  the  course. 


THE   HUMAN   SIGNIFICANCE   OF 
MATHEMATICS  l 

Homo  sum;  humani  nil  a  me  alienum  puto. 

—  TERENCE 

THE  subject  of  this  address  is  not  of  my  choosing.  It 
came  to  me  by  assignment.  I  may,  therefore,  be  allowed 
to  say  that  it  is  in  my  judgment  ideally  suited  to  the 
occasion.  This  meeting  is  held  here  upon  this  beautiful 
coast  because  of  the  presence  of  an  international  exposi- 
tion, and  we  are  thus  invited  to  a  befitting  largeness  and 
liberality  of  spirit.  An  international  exposition  prop- 
erly may  and  necessarily  will  admit  many  things  of  a 
character  too  technical  to  be  intelligible  to  any  one  but 
the  expert  and  the  specialist.  Such  things,  however, 
are  only  incidental  —  contributory,  indeed,  yet  inci- 
dental —  to  pursuit  of  the  principal  aim,  which  is,  I 
believe,  or  ought  to  -be,  the  representation  of  human 
things  as  human  —  an  exhibition  and  interpretation 
of  industries,  institutions,  sciences  and  arts,  not  pri- 
marily in  their  accidental  or  particular  character  as 
illustrating  individuals  or  classes  or  specific  localities 
or  times,  but  primarily  in  their  essential  and  universal 
character  as  representative  of  man.  A  world-exposition 
will,  therefore,  as  far  as  practicable,  avoid  placing  in 
the  forefront  matters  so  abstruse  as  to  be  fit  for  the 

1  An  address  delivered  August  3,  1915,  Berkeley,  Calif.,  at  a  joint  meet- 
ing of  the  American  Mathematical  Society,  the  American  Astronomical 
Society,  and  Section  A  of  the  American  Association  for  the  Advancement 
of  Science.  Printed  in  Science,  November  12,  1915. 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  27 

contemplation  and  understanding  of  none  but  special- 
ists; it  will,  as  a  whole,  and  in  all  its  principal  parts, 
address  itself  to  the  general  intelligence;  for  it  aims  at 
being,  for  the  multitudes  of  men  and  women  who  avail 
themselves  of  its  exhibitions  and  lessons,  an  exposition 
of  humanity:  an  exposition,  no  doubt,  of  the  activities 
and  aspirations  and  prowess  of  individual  men  and 
women,  but  of  men  and  women,  not  in  their  capacity, 
as  individuals,  but  as  representatives  of  humankind.  In- 
dividual achievements  are  not  the  object,  they  are  the 
means,  of  the  exposition.  The  object  is  humanity. 

What  is  the  human  significance  —  what  is  the  sig- 
nificance for  humanity  —  of  "the  mother  of  the  sci- 
ences"? And  how  may  the  matter  be  best  set  forth, 
not  for  the  special  advantage  of  professional  mathe- 
maticians, for  I  shall  take  the  liberty  of  having  these 
but  little  in  mind,  but  for  the  advantage  and  under- 
standing of  educated  men  and  women  in  general?  I  am 
unable  to  imagine  a  more  difficult  undertaking,  so  tech- 
nical, especially  in  its  language,  and  so  immense  is  the 
subject.  It  is  clear  that  the  task  is  far  beyond  the 
resources  of  an  hour's  discourse,  and  so  it  is  necessary 
to  restrict  and  select.  This  being  the  case,  what  is  it 
best  to  choose?  The  material  is  superabundant.  What 
part  of  it  or  aspect  of  it  is  most  available  for  the  end  in 
view?  "In  abundant  matter  to  speak  a  little  with 
elegance,"  says  Pindar,  "is  a  thing  for  the  wise  to  listen 
to."  It  is  not,  however,  a  question  of  elegance.  It  is 
a  question  of  emphasis,  of  clarity,  of  effectiveness.  What 
shall  be  our  major  theme? 

Shall  it  be  the  history  of  the  subject?  Shall  it  be 
the  modern  developments  of  mathematics,  its  present 
status  and  its  future  outlook?  Shall  it  be  the  utilities 
of  the  science,  its  so-called  applications,  its  service  in 


28  HUMAN    SIGNIFICANCE    OF    MATHEMATICS 

practical  affairs,  in  engineering  and  in  what  it  is  cus- 
tomary to  call  the  sciences  of  nature?  Shall  it  be  the 
logical  foundations  of  mathematics,  its  basic  principles, 
its  inner  nature,  its  characteristic  processes  and  struc- 
ture, the  differences  and  similitudes  that  come  to  light 
in  comparing  it  with  other  forms  of  scientific  and  philo- 
sophic activity?  Shall  it  be  the  bearings  of  the  science 
as  distinguished  from  its  applications  —  the  bearings  of 
it  as  a  spiritual  enterprise  upon  the  higher  concerns  of 
man  as  man?  It  might  be  any  one  of  these  things. 
They  are  all  of  them  great  and  inspiring  themes. 

It  is  easy  to  understand  that  a  historian  would  choose 
the  first.  The  history  of  mathematics  is  indeed  im- 
pressive, but  is  it  not  too  long  and  too  technical?  And 
is  it  not  already  accessible  in  a  large  published  litera- 
ture of  its  own?  I  grant,  the  historian  would  say,  that 
its  history  is  long,  for  in  respect  of  antiquity  mathematics 
is  a  rival  of  art,  surpassing  nearly  all  branches  of  sci- 
ence and  by  none  of  them  surpassed.  I  grant  that,  for 
laymen,  the  history  is  technical,  frightfully  technical, 
requiring  interpretation  in  the  interest  of  general  in- 
telligence. I  grant,  too,  that  the  history  owns  a  large 
literature,  but  this,  the  historian  would  say,  is  not 
designed  for  the  general  reader,  however  intelligent,  the 
numerous  minor  works  no  less  than  the  major  ones, 
including  that  culminating  monumental  work  of  Moritz 
Cantor,  being,  all  of  them,  addressed  to  specialists  and 
intelligible  to  them  alone.  And  yet  it  would  be  pos- 
sible to  tell  in  one  hour,  not  indeed  the  history  of  mathe- 
matics, but  a  true  story  of  it  that  would  be  intelligible 
to  all  and  would  show  its  human  significance  to  be 
profound,  manifold,  and  even  romantic.  It  would  be 
possible  to  show  historically  that  this  science,  which 
now  carries  its  head  so  high  in  the  tenuous  atmosphere 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  29 

of  pure  abstractions,  has  always  kept  its  feet  upon  the 
solid  earth;  it  would  be  possible  to  show  that  it  owns 
indeed  a  lowly  origin,  in  the  familiar  needs  of  common 
life,  in  the  homely  necessities  of  counting  herds  and 
measuring  lands;  it  would  be  possible  to  show  that, 
notwithstanding  its  birth  in  the  concrete  things  of  sense 
and  raw  reality,  it  yet  so  appealed  to  sheer  intellect  - 
and  we  must  not  forget  that  creative  intellect  is  the 
human  faculty  par  excellence  —  it  so  appealed  to  this 
distinctive  and  disinterested  faculty  of  man  that,  long 
before  the  science  rose  to  the  level  of  a  fine  art  in 
the  great  days  of  Euclid  and  Archimedes,  Plato  in  the 
wisdom  of  his  maturer  years  judged  it  essential  to  the 
education  of  freemen  because,  said  he,  there  is  in  it  a 
necessary  something  against  which  even  God  can  not 
contend  and  without  which  neither  gods  nor  demi-gods 
can  wisely  govern  mankind;  it  would  be  possible,  our 
historian  could  say,  to  show  historically  to  educated 
laymen  that,  even  prior  to  the  inventions  of  analytical 
geometry  and  the  infinitesimal  calculus,  mathematics 
had  played  an  indispensable  role  in  the  "Two  New 
Sciences"  of  physics  and  mechanics  in  which  Galileo 
laid  the  foundations  of  our  modern  knowledge  of  nature; 
it  would  be  possible  to  show  not  only  that  the  analytical 
geometry  of  Descartes  and  Fermat  and  the  calculus 
of  Leibnitz  and  Newton  have  been  and  are  essential 
to  our  still  advancing  conquest  of  the  sea,  but  that  it 
is  owing  to  the  power  of  these  instruments  that  the 
genius  of  such  as  Newton,  Laplace  and  Lagrange  has 
been  enabled  to  create  for  us  a  new  earth  and  a  new 
heavens  compared  with  which  the  Mosaic  cosmogony  or 
the  sublimest  creation  of  the  Greek  imagination  is  but 
"as  a  cabinet  of  brilliants,  or  rather  a  little  jewelled 
cup  found  in  the  ocean  or  the  wilderness";  it  would 


30  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

be  possible  to  show  historically  that,  just  because  the 
pursuit  of  mathematical  truth  has  been  for  the  most  part 
disinterested  —  led,  that  is,  by  wonder,  as  Aristotle  says, 
and  sustained  by  the  love  of  beauty  with  the  joy  of 
discovery  —  it  would  be  possible  to  show  that,  just 
because  of  the  disinterestedness  of  mathematical  re- 
search, this  science  has  been  so  well  prepared  to  meet 
everywhere  and  always,  as  they  have  arisen,  the  mathe- 
matical exigencies  of  natural  science  and  engineering; 
above  all,  it  would  be  feasible  to  show  historically  that 
to  the  same  disinterestedness  of  motive  operating  through 
the  centuries  we  owe  the  upbuilding  of  a  body  of  pure 
doctrine  so  towering  to-day  and  vast  that  no  man,  even 
though  he  have  the  "Andean  intellect"  of  a  Poincare, 
can  embrace  it  all.  This  much,  I  believe,  and  perhaps 
more,  touching  the  human  significance  of  mathematics, 
a  historian  of  the  science  might  reasonably  hope  to 
demonstrate  in  one  hour. 

More  difficult,  far  more  difficult,  I  think,  would  be 
the  task  of  a  pure  mathematician  who  aimed  at  an 
equivalent  result  by  expounding,  or  rather  by  delineat- 
ing, for  he  could  not  in  one  hour  so  much  as  begin  to 
expound,  the  modern  developments  of  the  subject. 
Could  he  contrive  even  to  delineate  them  in  a  way  to 
reveal  their  relation  to  what  is  essentially  humane?  Do 
but  consider  for  a  moment  the  nature  of  such  an  enter- 
prise. Mathematics  may  be  legitimately  pursued  for 
its  own  sake  or  for  the  sake  of  its  applications  or  with 
a  view  to  understanding  its  logical  foundations  and 
internal  structure  or  in  the  interest  of  magnanimity 
or  for  the  sake  of  its  bearings  upon  the  supreme  con- 
cerns of  man  as  man  or  from  two  or  more  of  these 
motives  combined.  Our  supposed  delineator  is  actuated 
by  the  first  of  them:  his  interest  in  mathematics  is  an 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  3! 

interest  in  mathematics  for  the  sake  of  mathematics; 
for  him  the  science  is  simply  a  large  and  growing  body 
of  logical  consistencies  or  compatibilities;  he  derives 
his  inspiration  from  the  muse  of  intellectual  harmony; 
he  is  a  pure  mathematician.  He  knows  that  pure  mathe- 
matics is  a  house  of  many  chambers;  he  knows  that 
its  foundations  lie  far  beneath  the  level  of  common 
thought;  and  that  the  superstructure,  quickly  tran- 
scending the  power  of  imagination  to  follow  it,  ascends 
higher  and  higher,  ever  keeping  open  to  the  sky;  he 
knows  that  the  manifold  chambers  —  each  of  them  a 
mansion  in  itself  —  are  all  of  them  connected  in  won- 
drous ways,  together  constituting  a  fit  laboratory  and 
dwelling  for  the  spirit  of  men  of  genius.  He  has  assumed 
the  task  of  presenting  a  vision  of  it  that  shall  be  worthy 
of  a  world-exposition.  Can  he  keep  the  obligation? 
He  wishes  to  show  that  the  life  and  work  of  pure 
mathematicians  are  human  life  and  work:  he  desires 
to  show  that  these  toilers  and  dwellers  in  the  chambers 
of  pure  thought  are  representative  men.  He  would 
exhibit  the  many-chambered  house  to  the  thronging 
multitudes  of  his  fellow  men  and  women;  he  would 
lead  them  into  it;  he  would  conduct  them  from  chamber 
to  chamber  by  the  curiously  winding  corridors,  passing 
now  downward,  now  upward,  by  delicate  passage- 
ways and  subtle  stairs;  he  would  show  them  that  the 
wondrous  castle  is  not  a  dead  or  static  affair  like  a 
structure  of  marble  or  steel,  but  a  living  architecture,  a 
living  mansion  of  life,  human  as  their  own;  he  would 
show  them  the  mathetic  spirit  at  work,  how  it  is  ever 
weaving,  tirelessly  weaving,  fabrics  of  beauty,  finer  than 
gossamer  yet  stronger  than  cables  of  steel;  he  would 
show  them  how  it  is  ever  enlarging  its  habitation,  deep- 
ening its  foundations,  expanding  more  and  more  and 


32  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

elevating  the  superstructure;  and,  what  is  even  more 
amazing,  how  it  perpetually  performs  the  curious  miracle 
of  permanence  combined  with  change,  transforming, 
that  is,  the  older  portions  of  the  edifice  without  destroy- 
ing it,  for  the  structure  is  eternal:  in  a  word,  he  would 
show  them  a  vision  of  the  whole,  and  he  would  do  it  in 
a  way  to  make  them  perceive  and  feel  that,  in  thus 
beholding  there  a  partial  and  progressive  attainment  of 
the  higher  ideals  of  man,  they  were  but  gazing  upon  a 
partial  and  progressive  realization  of  their  own  appe- 
titions  and  dreams. 

That  is  what  he  would  do.  But  how?  Mengenlehre, 
Zahlenlehre,  algebras  of  many  kinds,  countless  geometries 
of  countless  infinite  spaces,  function  theories,  trans- 
formations, invariants,  groups  and  the  rest  —  how  can 
these  with  all  their  structural  finesse,  with  their  heights 
and  depths  and  limitless  ramifications,  with  their  laby- 
rinthine and  interlocking  modern  developments  —  I  will 
not  say  how  can  they  be  presented  in  the  measure  and 
scale  of  a  great  exposition  —  but  how  is  it  possible  in 
one  hour  to  give  laymen  even  a  glimpse  of  the  endless 
array?  Nothing  could  be  more  extravagant  or  more 
absurd  than  such  an  undertaking.  Compared  with  it, 
the  American  traveler's  hope  of  being  able  to  see  Rome 
in  a  single  forenoon  was  a  most  reasonable  expectation. 
But  it  is  worth  while  trying  to  realize  how  stupendous 
the  absurdity  is. 

It  is  evident  that  our  would-be  delineator  must  com- 
promise. He  can  not  expound,  he  can  not  exhibit,  he 
can  not  even  delineate  the  doctrines  whose  human 
worth  he  would  thus  disclose  to  his  fellow  men  and 
women.  The  fault  is  neither  his  nor  theirs.  It  must 
be  imputed  to  the  nature  of  things.  But  he  need  not, 
therefore,  despair  and  he  need  not  surrender.  The 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  33 

method  he  has  proposed  —  the  method  of  exposition  — 
that  indeed  he  must  abandon  as  hopeless,  but  not  his 
aim.  He  is  addressing  men  and  women  who  are  no 
doubt  without  his  special  knowledge  and  his  special 
discipline,  as  he  in  his  turn  is  without  theirs,  but  who 
are  yet  essentially  like  himself.  He  would  have  them 
as  fellows  and  comrades  persuaded  of  the  dignity  of  his 
Fach:  he  would  have  them  feel  that  it  is  also  theirs; 
he  would  have  them  convinced  that  mathematics  stands 
for  an  immense  body  of  human  achievements,  for  a 
diversified  continent  of  pure  doctrine,  for  a  discovered 
world  of  intellectual  harmonies.  He  can  not  show  it 
to  them  as  a  painter  displays  a  canvas  or  as  an  architect 
presents  a  cathedral.  He  can  not  give  them  an  imme- 
diate vision  of  it,  but  he  can  give  them  intimations; 
by  appealing  to  their  fantasie  and,  through  analogy  with 
what  they  know,  to  their  understanding,  not  only  can 
he  convince  them  that  his  world  exists,  but  he  can  give 
them  an  intuitive  apprehension  of  its  living  presence 
and  its  meaning  for  humankind.  This  is  possible  be- 
cause, like  him,  they,  too,  are  idealists,  dreamers  and 
poets  —  such  essentially  are  all  men  and  women.  His 
auditors  or  his  readers  have  all  had  some  experience  of 
ideas  and  of  truth,  they  have  all  had  inklings  of  more 
beyond,  they  have  all  been  visited  and  quickened  by  a 
sense  of  the  limitless  possibilities  of  further  knowledge 
in  every  direction,  they  have  all  dreamed  of  the  perfect 
and  have  felt  its  lure.  They  are  thus  aware  that  the 
small  implies  the  large;  having  seen  hills,  they  can 
believe  in  mountains;  they  know  that  Euripides,  Shake- 
speare, Dante,  Goethe,  are  but  fulfillments  of  prophecies 
heard  in  peasant  tales  and  songs;  they  know  that  the 
symphonies  of  Beethoven  or  the  dramas  of  Wagner  are 
harbingered  in  the  melodies  and  the  sighs  of  those  who 


34  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

garner  grain  and  in  their  hearts  respond  to  the  music  of 
the  winds  or  the  "solemn  anthems  of  the  sea";  they 
sense  the  secret  by  which  the  astronomy  of  Newton  and 
Laplace  is  foretokened  in  the  shepherd's  watching  of 
the  stars;  and  knowing  thus  this  plain  spiritual  law 
of  progressiveness  and  implication,  they  are  prepared 
to  grasp  the  truth  that  modern  mathematics,  though 
they  do  not  understand  it,  is,  like  the  other  great  things, 
but  a  sublime  fulfillment,  the  realization  of  prophecies 
involved  in  what  they  themselves,  in  common  with 
other  educated  folk,  know  of  the  rudiments  of  the  sci- 
ence. Indeed,  they  would  marvel  if  upon  reflection 
it  did  not  seem  to  be  so.  Our  pure  mathematician  in 
speaking  to  his  fellow  men  and  women  of  his  science 
will  have  no  difficulty  in  persuading  them  that  he  is 
speaking  of  a  subject  immense  and  eternal.  As  born 
idealists  they  have  intimations  of  their  own  —  the 
evidence  of  intuition,  if  you  please  —  or  a  kind  of  insight 
resembling  that  of  the  mystic  —  that  in  the  world  of 
mind  there  must  be  something  deeper  and  higher,  stabler 
and  more  significant,  than  the  pitiful  ideas  in  life's 
routine  and  the  familiar  vocations  of  men.  They  are 
thus  prepared  to  believe,  before  they  are  told,  that 
behind  the  veil  there  exists  a  universum  of  exact  thought, 
an  everlasting  cosmos  of  ordered  ideas,  a  stable  world 
of  concatenated  truth.  In  their  study  of  the  elements, 
in  school  or  college,  they  may  have  caught  a  shimmer 
of  it  or,  in  rare  moments  of  illumination,  even  a  gleam. 
Of  the  existence,  the  reality,  the  actuality,  of  our  pure 
mathematician's  world  they  will  have  no  doubt,  and 
they  will  have  no  doubt  of  its  grandeur.  They  may 
even,  in  a  vague  way,  magnify  it  overmuch,  feeling  that 
it  is,  in  some  wise,  more  than  human,  significant  only 
for  the  rarely  gifted  spirit  that  dwells,  like  a  star, 


HUMAN   SIGNIFICANCE  OF   MATHEMATICS  35 

apart.  The  pure  mathematician's  difficulty  lies  in 
showing,  in  his  way,  that  such  is  not  the  case.  For  he 
does  not  wish  to  adduce  utilities  and  applications.  He 
is  well  aware  of  these.  He  knows  that  if  he  "would 
tell  them  they  are  more  in  number  than  the  sands." 
Neither  does  he  despise  them  as  of  little  moment.  On 
the  contrary,  he  values  them  as  precious.  But  he  wishes 
to  do  his  subject  and  his  auditors  the  honor  of  speaking 
from  a  higher  level:  he  desires  to  vindicate  the  worth 
of  mathematics  on  the  ground  of  its  sheer  ideality,  on 
the  ground  of  its  intellectual  harmony,  on  the  ground 
of  its  beauty,  "free  from  the  gorgeous  trappings"  of 
sense,  pure,  austere,  supreme.  To  do  this,  which  ought, 
it  seems,  to  be  easy,  experience  has  shown  to  be  exceed- 
ingly difficult.  For  the  multitude  of  men  and  women, 
even  the  educated  multitude,  are  wont  to  cry, 

Such  knowledge  is  too  wonderful  for  me, 
It  is  too  high,  I  can  not  attain  unto  it, 

thus  meaning  to  imply,  What,  then,  or  where  is  its 
human  significance?  Their  voice  is  heard  in  the  chal- 
lenge once  put  to  me  by  the  brilliant  author  of  "East 
London  Visions."  What,  said  he,  can  be  the  human 
significance  of  "this  majestic  intellectual  cosmos  of 
yours,  towering  up  like  a  million-lustred  iceberg  into  the 
arctic  night,"  seeing  that,  among  mankind,  none  is 
permitted  to  behold  its  more  resplendent  wonders  save 
the  mathematician  alone?  What  response  will  our  pure 
mathematician  make  to  this  challenge?  Make,  I  mean, 
if  he  be  not  a  wholly  naive  devotee  of  his  science  and 
so  have  failed  to  reflect  upon  the  deeper  grounds  of  its 
justification.  He  may  say,  for  one  thing,  what  Pro- 
fessor Klein  said  on  a  similar  occasion: 

Apart  from  the  fact  that  pure  mathematics  can  not  be  supplanted  by 
anything  else  as  a  means  for  developing  the  purely  logical  faculties  of  the 


36  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

mind,  there  must  be  considered  here,  as  elsewhere,  the  necessity  of  the 
presence  of  a  few  individuals  in  each  country  developed  in  far  higher  degree 
than  the  rest,  for  the  purpose  of  keeping  up  and  gradually  raising  the  general 
standard.  Even  a  slight  raising  of  the  general  level  can  be  accomplished 
only  when  some  few  minds  have  progressed  far  ahead  of  the  average. 

That  is  doubtless  a  weighty  consideration.  But  is 
it  all  or  the  best  that  may  be  said?  It  is  just  and 
important  but  it  does  not  go  far  enough;  it  is  not,  I 
fear,  very  convincing;  it  is  wanting  in  pungence  and 
edge;  it  does  not  touch  the  central  nerve  of  the  chal- 
lenge. Our  pure  mathematician  must  rally  his  sceptics 
with  sharper  considerations.  He  may  say  to  them: 
You  challenge  the  human  significance  of  the  higher 
developments  of  pure  mathematics  because  they  are 
inaccessible  to  all  but  a  few,  because  their  charm  is 
esoteric,  because  their  deeper  beauty  is  hid  from  nearly 
all  mankind.  Does  that  consideration  justify  your 
challenge?  You  are  individuals,  but  you  are  also 
members  of  a  race.  Have  you  as  individuals  no  human 
interest  nor  human  pride  in  the  highest  achievements 
of  your  race?  Is  nothing  human,  is  nothing  humane, 
except  mediocrity  and  the  commonplace?  Was  Phidias 
or  Michel  Angelo  less  human  than  the  carver  and 
painter  of  a  totem-pole?  Was  Euclid  or  Gauss  or 
Poincare  less  representative  of  man  than  the  countless 
millions  for  whom  mathematics  has  meant  only  the 
arithmetic  of  the  market  place  or  the  rude  geometry 
of  the  carpenter?  Does  the  quality  of  humanity  in 
human  thoughts  and  deeds  decrease  as  they  ascend 
towards  the  peaks  of  achievement,  and  increase  in 
proportion  as  they  become  vulgar,  attaining  an  upper 
limit  in  the  beasts?  Do  you  not  know  that  precisely 
the  reverse  is  true?  Do  you  not  count  aspiration  hu- 
mane? Do  you  not  see  that  it  is  not  the  common 
things  that  every  one  may  reach,  but  excellences  high- 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  37 

dwelling  among  the  rocks  —  do  you  not  know  that,  in 
respect  of  human  worth,  these  things,  which  but  few 
can  attain,  are  second  only  to  the  supreme  ideals  attain- 
able by  none? 

How  very  different  and  how  very  much  easier  the 
task  of  one  who  sought  to  vindicate  the  human  sig- 
nificance of  mathematics  on  the  ground  of  its  applica- 
tions! In  respect  of  temperamental  interest,  of  attitude 
and  outlook,  the  difference  between  the  pure  and  the 
applied  mathematician  is  profound.  It  is  —  if  we  may 
liken  spiritual  things  to  things  of  sense  —  much  like 
the  difference  between  one  who  greets  a  new-born  day 
because  of  its  glory  and  one  who  regards  it  as  a  time 
for  doing  chores  and  values  its  light  only  as  showing 
the  way.  For  the  former,  mathematics  is  justified  by 
its  supreme  beauty;  for  the  latter,  by  its  manifold  use. 
But  are  the  two  kinds  of  value  essentially  incompatible? 
They  are  certainly  not.  The  difference  is  essentially 
a  difference  of  authority  —  a  difference,  that  is,  of 
worth,  of  elevation,  of  excellence.  The  pure  mathe- 
matician and  the  applied  mathematician  sometimes  may, 
indeed  they  not  infrequently  do,  dwell  together  har- 
moniously in  a  single  personality.  If  our  spokesman  be 
such  a  one  —  and  I  will  not  suppose  the  shame  of  having 
the  utilities  of  the  science  represented  on  such  an  occa- 
sion by  one  incapable  of  regarding  it  as  anything  but 
a  tool,  for  that  would  be  disgraceful  —  if,  then,  our 
spokesman  be  such  a  one  as  I  have  supposed,  he  might 
properly  begin  as  follows:  In  speaking  to  you  of  the 
applications  of  mathematics  I  would  not  have  you  sup- 
pose, ladies  and  gentlemen,  that  I  am  thus  presenting 
the  highest  claims  of  the  science  to  your  regard;  for  its 
highest  justification  is  the  charm  of  its  immanent 
beauty;  I  do  hot  mean,  he  will  say,  the  beauty  of  ap- 


38  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

pearances  —  the  fleeting  beauties  of  sense,  though 
these,  too,  are  precious  —  even  the  outer  garment,  the 
changeful  robe,  of  reality  is  a  lovely  thing;  I  mean  the 
eternal  beauty  of  the  world  of  pure  thought;  I  mean 
intellectual  beauty;  in  mathematics  this  nearly  attains 
perfection;  and  "intellectual  beauty  is  self-sufficing "; 
uses,  on  the  other  hand,  are  not;  they  wear  an  aspect 
of  apology;  uses  resemble  excuses,  they  savor  a  little  of 
a  plea  in  mitigation.  Do  you  ask:  Why,  then,  plead 
them?  Because,  he  will  say,  many  good  people  have 
a  natural  incapacity  to  appreciate  anything  else;  be- 
cause, also,  many  of  the  applications,  especially  the 
higher  ones,  are  themselves  matters  of  exceeding  beauty; 
and  especially  because  I  wish  to  show,  not  only  that 
use  and  beauty  are  compatible  forms  of  worth,  but 
that  the  more  mathematics  has  been  cultivated  for  the 
sake  of  its  inner  charm,  the  fitter  has  it  become  for 
external  service. 

Having  thus  at  the  outset  put  himself  in  proper  light 
and  given  his  auditors  a  scholar's  warning  against  what 
would  else,  he  fears,  foster  a  disproportionment  of 
values,  what  will  he  go  on  to  signalize  among  the  utili- 
ties of  a  science  whose  primary  allegiance  to  logical 
rectitude  allies  it  to  art,  and  which  only  incidentally 
and  secondarily  shapes  itself  to  the  ends  of  instrumental 
service?  He  knows  that  the  applications  of  mathe- 
matics, if  one  will  but  trace  them  out  in  their  multi- 
farious ramifications,  are  as  many-sided  as  the  industries 
and  as  manifold  as  the  sciences  of  men,  penetrating 
everywhere  throughout  the  full  round  of  life.  What 
will  he  select?  He  will  not  dwell  long  upon  its  homely 
uses  in  the  rude  computations  and  mensurations  of 
counting-house  and  shop  and  factory  and  field,  for  this 
indispensable  yet  humble  manner  of  world-wide  and 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  39 

perpetual  service  is  known  of  all  men  and  women.  He 
will  quickly  pass  to  higher  considerations  —  to  naviga- 
tion, to  the  designing  of  ships,  to  the  surveying  of 
lands  and  seas,  and  the  charting  of  the  world,  to  the 
construction  of  reservoirs  and  aqueducts,  canals,  tunnels 
and  railroads,  to  the  modern  miracles  of  the  marine 
cable,  the  telegraph,  the  telephone,  to  the  multiform 
achievements  of  every  manner  of  modern  engineering, 
civil,  mechanical,  mining,  electrical,  by  which,  through 
the  advancing  conquest  of  land  and  sea  and  air  and 
space  and  time,  the  conveniences  and  the  prowess  of 
man  have  been  multiplied  a  billionfold.  It  need  not  be 
said  that  not  all  this  has  been  done  by  mathematics 
alone.  Far  from  it.  It  is,  of  course,  the  joint  achieve- 
ment of  many  sciences  and  arts,  but  —  and  just  this 
is  the  point  —  the  contributions  of  mathematics  to  the 
great  work,  direct  and  indirect,  have  been  indispensable. 
And  it  will  require  no  great  skill  in  our  speaker  to  show 
to  his  audience,  if  it  have  a  little  imagination,  that,  as  I 
have  said  elsewhere,  if  all  these  mathematical  contri- 
butions were  by  some  strange  spiritual  cataclysm  to  be 
suddenly  withdrawn,  the  life  and  body  of  industry  and 
commerce  would  suddenly  collapse  as  by  a  paralytic 
stroke,  the  now  splendid  outer  tokens  of  material  civiliza- 
tion would  quickly  perish,  and  the  face  of  our  planet 
would  at  once  assume  the  aspect  of  a  ruined  and  bank- 
rupt world.  For  such  is  the  amazing  utility,  such  the 
wealth  of  by-products,  if  you  please,  that  come  from 
a  science  and  art  that  owes  its  life,  its  continuity  and 
its  power  to  man's  love  of  intellectual  harmony  and 
pleads  its  inner  charm  as  its  sole  appropriate  justifica- 
tion. Indeed  it  appears  —  contrary  to  popular  belief  — 
that  in  our  world  there  is  nothing  else  quite  so  practical 
as  the  inspiration  of  a  muse. 


40  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

But  this  is  not  all  nor  nearly  all  to  which  our  applied 
mathematician  will  wish  to  invite  attention.  It  is  only 
the  beginning  of  it.  Even  if  he  does  not  allude  to  the 
quiet  service  continuously  and  everywhere  rendered  by 
mathematics  in  its  role  as  a  norm  or  standard  or  ideal 
in  every  field  of  thought  whether  exact  or  inexact,  he 
will  yet  desire  to  instance  forms  and  modes  of  applica- 
tion compared  with  which  those  we  have  mentioned, 
splendid  and  impressive  as  they  are,  are  meager  and 
mean.  For  those  we  have  mentioned  are  but  the  more 
obvious  applications  —  those,  namely,  that  continually 
announce  themselves  to  our  senses  everywhere  in  the 
affairs,  both  great  and  small,  of  the  workaday  world. 
But  the  really  great  applications  of  mathematics  —  those 
which,  rightly  understood,  best  of  all  demonstrate  the 
human  significance  of  the  science  —  are  not  thus  obvious; 
they  do  not,  like  the  others,  proclaim  themselves  in  the 
form  of  visible  facilities  and  visible  expedients  every- 
where in  the  offices,  the  shops,  and  the  highways  of 
commerce  and  industry;  they  are,  on  the  contrary, 
almost  as  abstract  and  esoteric  as  mathematics  itself, 
for  they  are  the  uses  and  applications  of  this  science 
in  other  sciences,  especially  in  astronomy,  in  mechanics 
and  in  physics,  but  also  and  increasingly  in  the  newer 
sciences  of  chemistry,  geology,  mineralogy,  botany, 
zoology,  economics,  statistics  and  even  psychology,  not 
to  mention  the  great  science  and  art  of  architecture. 
In  the  matter  of  exhibiting  the  endless  and  intricate 
applications  of  mathematics  to  the  natural  sciences, 
applications  ranging  from  the  plainest  facts  of  crystal- 
lography to  the  faint  bearings  of  the  kinetic  theory  of 
gases  upon  the  constitution  of  the  Milky  Way,  our 
speaker's  task  is  quite  as  hopeless  as  we  found  the  pure 
mathematician's  to  be;  and  he,  too,  will  have  to  com- 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  4! 

promise;  he  will  have  to  request  his  auditors  to  ac- 
quaint themselves  at  their  leisure  with  the  available 
literature  of  the  subject  and  especially  to  read  atten- 
tively the  great  work  of  John  Theodore  Merz  dealing 
with  the  "History  of  European  Thought  in  the  Nine- 
teenth Century,"  where  they  will  find,  in  a  form  fit  for 
the  general  reader,  how  central  has  been  the  r61e  of 
mathematics  in  all  the  principal  attempts  of  natural 
science  to  find  a  cosmos  in  the  seeming  chaos  of  the 
natural  world.  Another  many-sided  work  that  in  this 
connection  he  may  wish  to  commend  as  being  in  large 
part  intelligible  to  men  and  women  of  general  education 
and  catholic  mind  is  Enriques's  "Problems  of  Science." 

I  turn  now  for  a  moment  to  the  prospects  of  one  who 
might  choose  to  devote  the  hour  to  an  exposition  or 
an  indication  of  modern  developments  in  what  it  is 
customary  to  call  the  foundations  of  mathematics  —  to 
a  characterization,  that  is,  and  estimate  of  that  far- 
reaching  and  still  advancing  critical  movement  which 
has  to  do  with  the  relations  of  the  science,  philosophi- 
cally considered,  to  the  sciences  of  logic  and  methodology. 
What  can  he  say  on  this  great  theme  that  will  be  in- 
telligible and  edifying  to  the  multitudes  of  men  and 
women  who,  though  mathematically  inexpert,  yet  have 
a  genuine  humane  curiosity  respecting  even  the  pro- 
founder  and  subtler  life  and  achievements  of  science? 
He  can  point  out  that  mathematics,  like  all  the  other 
sciences,  like  the  arts  too,  for  that  matter,  and  like 
philosophy,  originates  in  the  refining  process  of  reflec- 
tion upon  the  crude  data  of  common  sense ;  he  can  point 
out  that  this  process  has  gradually  yielded  from  out  the 
raw  material  and  still  continues  to  yield  more  and  more 
ideas  of  approximate  perfection  in  the  respects  of  pre- 
cision and  form;  he  can  point  out  that  such  ideas,  thus 


42  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

disentangled  and  trimmed  of  their  native  vagueness  and 
indetermination,  disclose  their  mutual  relationships  and 
so  become  amenable  to  the  concatenative  processes  of 
logic;  and  he  can  point  out  that  these  polished  ideas 
with  their  mutual  relationships  become  the  bases  or 
the  content  of  various  branches  of  mathematics,  which 
thus  tower  above  common  sense  and  appear  to  grow 
out  of  it  and  to  stand  upon  it  like  trees  or  forests 
upon  the  earth.  He  will  point  out,  however,  that  this 
appearance,  like  most  other  obvious  appearances,  is  de- 
ceiving; he  will,  that  is,  point  out  that  these  upward- 
growing  sciences  or  branches  of  science  are  found,  in 
the  light  of  further  reflection,  to  be  downward-growing 
as  well,  pushing  their  roots  deeper  and  deeper  into  a 
dark  soil  far  beneath  the  ground  of  evident  common 
sense;  indeed,  he  will  show  that  common  sense  is  thus, 
in  its  relation  to  mathematics,  but  as  a  sense-litten  mist 
enveloping  only  the  mid-portion  of  the  stately  structure, 
which,  like  a  towering  mountain,  at  once  ascends  into 
the  limpid  ether  far  above  the  shining  cloud  and  rests 
upon  a  base  of  subterranean  rock  far  below;  he  will 
point  out  that,  accordingly,  mathematicians,  in  respect 
of  temperamental  interest,  fall  into  two  classes  —  the 
class  of  those  who  cultivate  the  upward-growing  of  the 
science,  working  thus  in  the  upper  regions  of  clearer 
light,  and  the  class  of  those  who  devote  themselves  to 
exploring  the  deep-plunging  roots  of  the  science;  and  it 
is,  he  will  say,  to  the  critical  activity  of  the  latter  class 
—  the  logicians  and  philosophers  of  mathematics  —  that 
we  owe  the  discovery  of  what  we  are  wont  to  call  the 
foundations  of  mathematics  —  the  great  discovery,  that 
is,  of  an  immense  mathematical  stt&-structure,  which 
penetrates  far  beneath  the  stratum  of  common  sense 
and  of  which  many  of  even  the  greatest  mathematicians 


HUMAN   SIGNIFICANCE   OF  MATHEMATICS  43 

of  former  times  were  not  aware.  But  whilst  such  founda- 
tional  research  is  in  the  main  a  modern  phenomenon, 
it  is  by  no  means  exclusively  such;  and  to  protect  his 
auditors  against  a  false  perspective  in  this  regard  and 
the  peril  of  an  overweening  pride  in  the  achievements  of 
their  own  time,  our  speaker  may  recommend  to  them 
the  perusal  of  Thomas  L.  Heath's  superb  edition  of 
Euclid's  "Elements"  where,  especially  in  the  first  vol- 
ume, they  will  be  much  edified  to  find,  in  the  rich 
abundance  of  critical  citation  and  commentary  which 
the  translator  has  there  brought  together,  that  the  re- 
fined and  elaborate  logico-mathematical  researches  of  our 
own  time  have  been  only  a  deepening  and  widening  of 
the  keen  mathematical  criticism  of  a  few  centuries  im- 
mediately preceding  and  following  the  great  date  of 
Euclid.  Indeed  but  for  that  general  declension  of  Greek 
spirit  which  Professor  Gilbert  Murray  in  his  "Four 
Stages  of  Greek  Religion"  has  happily  characterized  as 
"the  failure  of  nerve,"  what  we  know  as  the  modern 
critical  movement  in  mathematics  might  well  have  come 
to  its  present  culmination,  so  far  at  least  as  pure  geom- 
etry is  concerned,  fifteen  hundred  or  more  years  ago. 
It  is  a  pity  that  the  deeper  and  stabler  things  of  science 
and  the  profounder  spirit  of  man  can  not  be  here 
disclosed  in  a  manner  commensurate  with  the  great 
exposition,  surrounding  us,  of  the  manifold  practical 
arts  and  industries  of  the  world.  It  is  a  pity  there  is 
no  means  by  which  our  speaker  might,  in  a  manner 
befitting  the  subject  and  the  occasion,  exhibit  intelligibly 
to  his  fellow  men  and  women  the  ways  and  results  of 
the  last  hundred  years  of  research  into  the  groundwork 
of  mathematical  science  and  therewith  the  highly  im- 
portant modern  developments  in  logic  and  the  theory 
of  knowledge.  How  astonished  the  beholders  would 


44  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

be,  how  delighted  too,  and  proud  to  belong  to  a  race 
capable  of  such  patience  and  toil,  of  such  disinterested 
devotion,  of  such  intellectual  finesse  and  depth  of  pene- 
tration. I  can  think  of  no  other  spectacle  quite  so  im- 
pressive as  the  inner  vision  of  all  the  manifold  branches 
of  rigorous  thought  seen  to  constitute  one  immense 
structure  of  autonomous  doctrine  reposing  upon  the 
spiritual  basis  of  a  few  select  ideas  and,  superior  to  the 
fading  beauties  of  time  and  sense,  shining  there  like  a 
celestial  city,  in  "the  white  radiance  of  eternity."  That 
is  the  vision  of  mathematics  that  a  student  of  its  phi- 
losophy would,  were  it  possible,  present  to  his  fellow 
men  and  women. 

In  view  of  the  foregoing  considerations  it  evidently  is, 
I  think,  in  the  nature  of  the  case  impossible  to  give  an 
adequate  sense  of  the  human  worth  of  mathematics  if 
one  choose  to  devote  the  hour  to  any  one  of  the  great 
aspects  of  it  with  which  we  have  been  thus  far  con- 
cerned. Neither  the  history  of  the  subject  nor  its 
present  estate  nor  its  applications  nor  its  logical  founda- 
tions —  no  one  of  these  themes  lends  itself  well  to  the 
purpose  of  such  exposition,  and  still  less  do  two  or  more 
of  them  combined.  Even  if  such  were  not  the  case  I 
should  yet  feel  bound  to  pursue  another  course;  for  I 
have  been  long  persuaded  that,  in  respect  of  its  human 
significance,  mathematics  invites  to  a  point  of  view 
which,  unless  I  am  mistaken,  has  not  been  taken  and 
held  in  former  attempts  at  appreciation.  I  have  al- 
ready alluded  to  bearings  of  mathematics  as  distin- 
guished from  applications.  It  is  with  its  bearings  that 
I  wish  to  deal.  I  mean  its  bearings  upon  the  higher 
concerns  of  man  as  man  —  those  interests,  namely, 
which  have  impelled  him  to  seek,  over  and  above  the 
needs  of  raiment  and  shelter  and  food,  some  inner 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  45 

adjustment  of  life  to  the  poignant  limitations  of  life  in 
our  world  and  which  have  thus  drawn  him  to  manifold 
forms  of  wisdom,  not  only  to  mathematics  and  natural 
science,  but  also  to  literature  and  philosophy,  to  religion 
and  art,  and  theories  of  righteousness.  What  is  the 
rfile  of  mathematics  in  this  perpetual  endeavor  of  the 
human  spirit  everywhere  to  win  reconciliation  of  its 
dreams  and  aspirations  with  the  baffling  conditions  and 
tragic  facts  of  life  and  the  world?  What  is  its  relation 
to  the  universal  quest  of  man  for  some  supreme  and 
abiding  good  that  shall  assuage  or  annul  the  discords 
and  tyrannies  of  time  and  limitation,  withholding  less 
and  less,  as  time  goes  by,  the  freedom  and  the  peace 
of  an  ideal  harmony  infinite  and  eternal? 

In  endeavoring  to  suggest,  in  the  time  remaining  for 
this  address,  a  partial  answer  to  that  great  question,  in 
attempting,  that  is,  to  indicate  the  relations  of  mathe- 
matics to  the  supreme  ideals  of  mankind,  it  will  be 
necessary  to  seek  a  perspective  point  of  view  and  to 
deal  with  large  matters  in  a  large  way. 

Of  the  countless  variety  of  appetitions  and  aspirations 
that  have  given  direction  and  aim  to  the  energies  of 
men  and  that,  together  with  the  constraining  conditions 
of  life  in  our  world,  have  shaped  the  course  and  deter- 
mined the  issues  of  human  history,  it  is  doubtless  not 
yet  possible  to  attempt  confident  and  thoroughgoing 
classification  according  to  the  principle  of  relative  dig- 
nity or  that  of  relative  strength.  If,  however,  we  ask 
whether,  in  the  great  throng  of  passional  determinants 
of  human  thought  and  life,  there  is  one  supreme  passion, 
one  that  in  varying  degrees  of  consciousness  controls 
the  rest,  unifying  the  spiritual  enterprises  of  our  race 
in  directing  and  converging  them  all  upon  a  single 
sovereign  aim,  the  answer,  I  believe,  can  not  be  doubt- 


46  HUMAN    SIGNIFICANCE   OF   MATHEMATICS 

ful:  the  activities  and  desires  of  mankind  are  indeed 
subject  to  such  imperial  direction  and  control.  And  if 
now  we  ask  what  the  sovereign  passion  is,  again  the 
answer  can  hardly  admit  of  question  or  doubt.  In  order 
to  see  even  a  priori  what  the  answer  must  be,  we  have 
only  to  imagine  a  race  of  beings  endowed  with  our 
human  craving  for  stability,  for  freedom,  and  for  per- 
petuity of  life  and  its  fleeting  goods,  we  have  only  to 
fancy  such  a  race  flung,  without  equipment  of  knowledge 
or  strength,  into  the  depths  of  a  treacherous  universe 
of  matter  and  force  where  they  are  tossed,  buffeted  and 
torn  by  the  tumultuous  onward-rushing  flood  of  the 
cosmic  stream,  originating  they  know  not  whence  and 
flowing  they  know  not  why  nor  whither,  we  have,  I  say, 
only  to  imagine  this,  sympathetically,  which  ought  to 
be  easy  for  us  as  men,  and  then  to  ask  ourselves  what 
would  naturally  be  the  controlling  passion  and  dominant 
enterprise  of  such  a  race  —  unless,  indeed,  we  suppose 
it  to  become  strangely  enamored  of  distress  or  to  be 
driven  by  despair  to  self -extinction.  We  humans  re- 
quire no  Gotama  nor  Heracleitus  to  tell  us  that  man's 
lot  is  cast  in  a  world  where  naught  abides.  The  uni- 
versal impermanence  -of  things,  the  inevitableness  of 
decay,  the  mocking  frustration  of  deepest  yearnings 
and  fondest  dreams,  all  this  has  been  keenly  realized 
wherever  men  and  women  have  had  seeing  eyes  or  been 
even  a  little  touched  with  the  malady  of  meditation, 
and  everywhere  in  the  literature  of  power  is  heard  the 
cry  of  the  mournful  truth.  "The  life  of  man,"  said 
the  Spirit  of  the  Ocean,  "passes  by  like  a  galloping 
horse,  changing  at  every  turn,  at  every  hour." 

"Great  treasure  halls  hath  Zeus  in  heaven, 
From  whence  to  man  strange  dooms  be  given, 
Past  hope  or  fear." 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  47 

Such  is  the  universal  note.  Whether  we  glance  at  the 
question  in  a  measure  a  priori,  as  above,  or  look  into 
the  cravings  of  our  own  hearts,  or  survey  the  history 
of  human  emotion  and  thought,  we  shall  find,  I  think, 
in  each  and  all  these  ways,  that  human  life  owns  the 
supremacy  of  one  desire:  it  is  the  passion  for  emancipa- 
tion, for  release  from  life's  limitations  and  the  tyranny 
of  change:  it  is  our  human  passion  for  some  ageless 
form  of  reality,  some  everlasting  vantage-ground  or  rock 
to  stand  upon,  some  haven  of  refuge  from  the  all- 
devouring  transformations  of  the  weltering  sea.  And 
so  it  is  that  our  human  aims,  aspirations,  and  toils 
thus  find  their  highest  unity  —  their  only  intelligible 
unity  —  in  the  spirit's  quest  of  a  stable  world,  in  its 
endless  search  for  some  mode  or  form  of  reality  that 
is  at  once  infinite,  changeless,  eternal. 

Does  some  one  say:  This  may  be  granted,  but  what 
is  the  point  of  it  all?  It  is  obviously  true  enough,  but 
what,  pray,  can  be  its  bearing  upon  the  matter  in  hand? 
What  light  does  it  throw  upon  the  human  significance 
of  mathematics?  The  question  is  timely  and  just.  The 
answer,  which  will  grow  in  fullness  and  clarity  as  we 
proceed,  may  be  at  once  begun. 

How  long  our  human  ancestors,  in  remote  ages,  may 
have  groped,  as  some  of  their  descendants  even  now 
grope,  among  the  things  of  sense,  in  the  hope  of  finding 
there  the  desiderated  good,  we  do  not  know  —  past  time 
is  long  and  the  evolution  of  wisdom  has  been  slow.  We 
do  know  that,  long  before  the  beginnings  of  recorded 
history,  superior  men  —  advanced  representatives  of 
their  kind  —  must  have  learned  that  the  deliverance 
sought  was  not  to  be  found  among  the  objects  of  the 
mobile  world,  and  so  the  spirit's  quest  passed  from 
thence;  passed  from  the  realm  of  perception  and  sense 


48  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

to  the  realm  of  concept  and  reason:  thought  ceased, 
that  is,  to  be  merely  the  unconscious  means  of  pursuit 
and  became  itself  the  quarry  —  mind  had  discovered 
mind;  and  there,  in  the  realm  of  ideas,  in  the  realm  of 
spirit  proper,  in  the  world  of  reason  or  thought,  the 
great  search  —  far  outrunning  historic  time  —  has  been 
endlessly  carried  on,  with  varying  fortunes,  indeed,  but 
without  despair  or  breach  of  continuity,  meanwhile 
multiplying  its  resources  and  assuming  gradually,  as  the 
years  and  centuries  have  passed,  the  characters  and 
forms  of  what  we  know  today  as  philosophy  and  science 
and  art.  I  have  mentioned  the  passing  of  the  quest  from 
the  realm  of  sense  to  the  realm  of  conception:  a  most 
notable  transition  in  the  career  of  mind  and  especially 
significant  for  the  view  I  am  aiming  to  sketch.  For 
thought,  in  thus  becoming  a  conscious  subject  or  object 
of  thought,  then  began  its  destined  course  in  reason: 
in  ceasing  to  be  merely  an  unconscious  means  of  pursuit 
and  becoming  itself  the  quarry,  it  definitely  entered 
upon  the  arduous  way  that  leads  to  the  goal  of  rigor. 
And  so  it  is  evident  that  the  way  in  question  is  not  a 
private  way;  it  does  not  belong  exclusively  to  mathe- 
matics; it  is  public  property;  it  is  the  highway  of  con- 
ceptual research.  For  it  is  a  mistake  to  imagine  that 
mathematics,  in  virtue  of  its  reputed  exactitude,  is  an 
insulated  science,  dwelling  apart  in  isolation  from  other 
forms  and  modes  of  conceptual  activity.  It  would  be 
such,  were  its  rigor  absolute;  for  between  a  perfection 
and  any  approximation  thereto,  however  close,  there 
always  remains  an  infinitude  of  steps.  But  the  rigor 
of  mathematics  is  not  absolute  —  absolute  rigor  is  an 
ideal,  to  be,  like  other  ideals,  aspired  unto,  forever 
approached,  but  never  quite  attained,  for  such  attain- 
ment would  mean  that  every  possibility  of  error  or 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  49 

inde termination,  however  slight,  had  been  eliminated 
from  idea,  from  symbol,  and  from  argumentation.  We 
know,  however,  that  such  elimination  can  never  be 
complete,  unless  indeed  the  human  mind  shall  one  day 
lose  its  insatiable  faculty  for  doubting.  What,  then, 
is  the  distinction  of  mathematics  on  the  score  of  exacti- 
tude? Its  distinction  lies,  not  in  the  attainment  of 
rigor  absolute,  but  partly  in  its  exceptional  devotion 
thereto  and  especially  in  the  advancement  it  has  made 
along  the  endless  path  that  leads  towards  that  perfec- 
tion. But,  as  I  have  already  said,  it  must  not  be 
thought  that  mathematics  is  the  sole  traveler  upon  the 
way.  It  is  important  to  see  clearly  that  it  is  far  from 
being  thus  a  solitary  enterprise.  First,  however,  let 
us  adjust  our  imagery  to  a  better  correspondence  with 
the  facts.  I  have  spoken  of  the  path.  We  know,  how- 
ever, that  the  paths  are  many,  as  many  as  the  varieties 
of  conceptual  subject-matter,  all  of  them  converging 
towards  the  same  high  goal.  We  see  them  originate 
here,  there  and  yonder  in  the  soil  and  haze  of  common 
thought;  we  see  how  indistinct  they  are  at  first  —  how 
ill-defined;  we  observe  how  they  improve  in  that  regard 
as  the  ideas  involved  grow  clearer  and  clearer,  more  and 
more  amenable  to  the  use  and  governance  of  logic.  At 
length,  when  thought,  in  its  progress  along  any  one  of 
the  many  courses,  has  reached  a  high  degree  of  refine- 
ment, precision  and  certitude,  then  and  thereafter,  but 
not  before,  we  call  it  mathematical  thought;  it  has 
undergone  a  long  process  of  refining  evolution  and 
acquired  at  length  the  name  of  mathematics;  it  is 
not,  however,  the  creature  of  its  name;  what  is  called 
mathematics  has  been  long  upon  the  way,  owning  at 
previous  stages  other  designations  —  common  sense, 
practical  art  perhaps,  speculation,  theology  it  may  be, 


5<D  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

philosophy,  natural  science,  or  it  may  be  for  many  a 
millennium  no  name  at  all.  Is  it,  then,  only  a  question 
of  names?  In  a  sense,  yes:  the  ideal  of  thought  is 
rigor;  mathematics  is  the  name  that  usage  employs  to 
designate,  not  attainment  of  the  ideal,  for  it  can  not 
be  attained,  but  its  devoted  pursuit  and  close  approxi- 
mation. But  this  is  not  the  essence  of  the  matter. 
The  essence  is  that  all  thought,  thought  in  all  its  stages, 
however  rude,  however  refined,  however  named,  owns 
the  unity  of  being  human:  spiritual  activities  are  one. 
Mathematics  thus  belongs  to  the  great  family  of  spiritual 
enterprises  of  man.  These  enterprises,  all  the  members 
of  the  great  family,  however  diverse  in  form,  in  modes 
of  life,  in  methods  of  toil,  in  their  progress  along  the 
way  that  leads  towards  logical  rectitude,  are  alike  chil- 
dren of  one  great  passion.  In  genesis,  in  spirit  and 
aspiration,  in  motive  and  aim,  natural  science,  theology, 
philosophy,  jurisprudence,  religion  and  art  are  one  with 
mathematics:  they  are  all  of  them  sprung  from  the 
human  spirit's  craving  for  invariant  reality  in  a  world 
of  tragic  change;  they  all  of  them  aim  at  rescuing  man 
from  "the  blind  hurry  of  the  universe  from  vanity  to 
vanity":  they  seek  cosmic  stability  —  a  world  of  abiding 
worth,  where  the  broken  promises  of  hope  shall  be 
healed  and  infinite  aspiration  shall  cease  to  be  mocked. 

Such  has  been  the  universal  and  dominant  aim  and 
such  are  the  cardinal  forms  that  time  has  given  its 
prosecution. 

And  now  we  must  ask:  What  have  been  the  fruits 
of  the  endless  toil?  What  has  the  high  emprize  won? 
And  what  especially,  have  been  the  contributions  of 
mathematics  to  the  total  gain?  To  recount  the  story 
of  the  spirit's  quest  for  ageless  forms  of  reality  would 
be  to  tell  afresh,  from  a  new  point  of  view,  the  his- 


HUMAN   SIGNIFICANCE   OF   MATHEMATICS  $1 

tory  of  human  thought,  so  many  and  so  diverse  are 
the  modes  or  aspects  of  being  that  men  have  found  or 
fancied  to  be  eternal.  Edifying  indeed  would  be  the 
tale,  but  it  is  long,  and  the  hour  contracts.  Even  a 
meager  delineation  is  hardly  possible  here.  Yet  we 
must  not  fail  to  glance  at  the  endless  array  and  to  call, 
at  least  in  part,  the  roll  of  major  things.  But  where 
begin?  Shall  it  be  in  theology?  How  memory  responds 
to  the  magic  word.  "The  past  rises  before  us  like  a 
dream."  As  the  long  succession  of  the  theological  cen- 
turies passes  by,  what  a  marvelous  pageant  do  they 
present  of  human  ideals,  contrivings  and  dreams,  both 
rational  and  superrational.  Alpha  and  Omega,  the  be- 
ginning and  ending,  which  is,  which  was  and  which  is 
to  come;  I  Am  That  I  Am;  Father  of  lights  with  which 
is  no  variableness,  neither  shadow  of  turning;  the 
bonitas,  unit  as.  infinitas,  immutabilitas  of  Deity;  the 
undying  principle  of  soul;  the  sublime  hierarchy  of 
immortal  angels,  terrific  and  precious,  discoursed  of  by 
sages,  commemorated  by  art,  feared  and  loved  by  mil- 
lions of  men  and  women  and  children:  these  things  may 
suffice  to  remind  us  of  the  invariant  forms  of  reality 
found  or  invented  by  theology  in  her  age-long  toil  and 
passion  to  conquer  the  mutations  of  time  by  means  of 
things  eternal. 

But  theology's  record  is  only  an  immense  chapter  of 
the  vastly  more  inclusive  annals  of  world-wide  philo- 
sophic speculation  running  through  the  ages.  If  we 
turn  to  philosophy  understood  in  the  larger  sense,  if  we 
ask  what  answers  she  has  made  in  the  long  course  of 
time  to  the  question  of  what  is  eternal,  so  diverse  and 
manifold  are  the  voices  heard  across  the  centuries,  from 
the  East  and  from  the  West,  that  the  combined  response 
must  needs  seem  to  an  unaccustomed  ear  like  an  infinite 


52  HUMAN   SIGNIFICANCE   OF   MATHEMATICS 

babel  of  tongues:  the  Confucian  Way  of  Heaven;  the 
mystic  Tao,  so  much  resembling  fate,  of  Lao  Tzu  and 
Chuang  Tzu;  Buddhism's  inexorable  spiritual  law  of 
cause  and  effect  and  its  everlasting  extinction  of  indi- 
viduality in  Nirvana  —  the  final  blowing  out  of  con- 
sciousness and  character  alike;  Ahura  Mazda,  the  holy 
One,  of  Zarathustra;  Fate,  especially  in  the  Greek 
tragedies  and  Greek  religion  —  the  chain  of  causes  in 
nature,  "the  compulsion  in  the  way  things  grow,"  a  fine 
thread  running  through  the  whole  of  existence  and 
binding  even  the  gods;  the  cosmic  matter,  or  TO 
aTT€Lpov  of  Anaximander;  the  cosmic  order,  the  rhythm 
of  events,  the  logos  or  reason  or  nous,  of  Heracleitus; 
the  finite,  space-filling  sphere,  or  One,  of  the  deep 
Parmenides;  the  four  material  and  two  psychic,  six 
eternal,  elements,  of  Empedocles;  the  infinitude  of  ever- 
lasting mind-moved  simple  substances  of  Anaxagoras; 
the  infinite  multitude  and  endless  variety  of  invariant 
"seeds  of  things"  of  Leucippus,  Democritus,  Epicurus 
and  Lucretius,  together  with  their  doctrines  of  absolute 
void  and  the  conservations  of  mass  and  motion  and 
infinite  room  or  space;  Plato's  eternal  world  of  pure 
ideas;  the  great  Cosmic  Year  of  a  thousand  thinkers, 
rolling  in  vast  endlessly  repeated  cycles  on  the  beginning- 
less,  endless  course  of  time  from  eternity  to  eternity; 
the  changeless  thought-forms  of  Zeno,  Gorgias  and  Aris- 
totle; Leibnitz's  indestructible,  pre-established  harmony; 
Spinoza's  infinite  unalterable  substance;  the  Absolute 
of  the  Hegelian  school;  and  so  on  and  on  far  beyond 
the  limits  of  practicable  enumeration.  This  somewhat 
random  partial  list  of  things  will  serve  to  recall  and  to 
represent  the  enormous  motley  crowd  of  answers  that 
the  ages  of  philosophic  speculation  have  made  to  the 
supreme  inquiry  of  the  human  spirit:  what  is  there 


HUMAN    SIGNIFICANCE    OF    MATHEMATICS  53 

that  survives  the  mutations  of  time,  abiding  unchanged 
despite  the  whirling  flux  of  life  and  the  world? 

And  now,  in  the  interest  of  further  representing  salient 
features  in  a  large  perspective  view,  let  me  next  ask 
what  contribution  to  the  solution  of  the  great  problem 
has  been  made  by  jurisprudence.  Jurisprudence  is  no 
doubt  at  once  a  branch  of  philosophy  and  a  branch  of 
science,  but  it  has  an  interest,  a  direction  and  a  char- 
acter of  its  own.  And  for  the  sake  of  due  emphasis  it 
will  be  well  worth  while  to  remind  ourselves  specifically 
of  the  half-forgotten  fact  that,  in  its  quest  for  justice 
and  order  among  men,  jurisprudence  long  ago  found  an 
answer  to  our  oft-stated  riddle  of  the  world,  an  answer 
which,  though  but  a  partial  one,  yet  satisfied  the  greatest 
thinkers  for  many  centuries,  and  which,  owing  to  the 
inborn  supernalizing  proclivity  of  the  human  mind,  still 
exercises  sway  over  the  thought  of  the  great  majority  of 
mankind.  I  allude  to  the  conception  of  jus  natural?  or 
lex  natunr.  the  doctrine  that  in  the  order  of  Nature 
there  somehow  exists  a  perfect,  invariant,  universally 
and  eternally  valid  system  or  prototype  of  law  over  and 
above  the  imperfect  laws  and  changeful  polities  of  men 
—  a  conception  and  doctrine  long  familiar  in  the  juristic 
thought  of  antiquity,  dominating,  for  example,  the  An- 
tigone of  Sophocles,  penetrating  the  Republic  and  the 
Laws  of  Plato,  proclaimed  by  Demosthenes  in  the  Ora- 
tion on  the  Crown,  becoming,  largely  through  the 
Republic  and  the  Laws  of  Cicero,  the  crowning  con- 
ception of  the  imperial  jurisprudence  of  Rome,  and  still 
holding  sway,  as  I  have  said,  except  in  the  case  of  our 
doubting  Thomases  of  the  law,  who  virtually  deny 
our  world  the  existence  of  any  perfection  whatever 
because  they  can  not,  so  to  speak,  feel  it  with  the 
hand,  as  if  they  did  not  know  that  to  suppose  an 


54  HUMAN    SIGNIFICANCE    OF    MATHEMATICS 

ideal  to  be  thus  realized  would  be  a  flat  contradiction 
in  terms. 

If  we  turn  for  a  moment  to  art  and  enquire  what  has 
been  her  relation  to  the  poignant  riddle,  shall  we  not 
thus  be  going  too  far  afield?  The  answer  is  certainly 
no.  In  (Eternitatem  pingo,  said  Zeuxis,  the  Greek  painter. 
"The  purpose  of  art,"  says  John  La  Farge,  "is  com- 
memoration." In  these  two  sayings,  one  of  them  ancient, 
the  other  modern,  we  have,  I  think,  the  evident  clue. 
They  do  but  tell  us  that  art,  like  the  other  great  enter- 
prises of  man,  springs  from  our  spirit's  coveting  of 
worth  that  abides.  Like  theology,  like  philosophy,  like 
jurisprudence,  like  natural  science,  too,  as  I  mean  to 
point  out  further,  and  like  mathematics,  art  is  born  of 
the  universal  passion  for  the  dignity  of  things  eternal. 
Her  quest,  like  theirs,  has  been  a  search  for  invariants, 
for  goods  that  are  everlasting.  And  what  has  she  found? 
The  answer  is  simple.  "The  idea  of  beauty  in  each 
species  of  being,"  said  Joshua  Reynolds,  "is  perfect, 
invariable,  divine."  We  know  that  by  a  faculty  of 
imaginative,  mystical,  idealizing  discernment  there  is 
revealed  to  us,  amid  the  fleeting  beauties  of  Time,  the 
immobile  presence  of  Eternal  beauty,  immutable  arche- 
type and  source  of  the  grace  and  loveliness  beheld  in  the 
shifting  scenes  of  the  flowing  world  of  sense.  Such,  I 
take  it,  is  art's  contribution  to  our  human  release  from 
the  tyranny  of  change  and  the  law  of  death. 

And  now  what  should  be  said  of  science?  Not  so 
brief  and  far  less  simple  would  be  the  task  of  character- 
izing or  even  enumerating  the  many  things  that  in  the 
great  drama  of  modern  science  have  been  assigned  the 
role  of  invariant  forms  of  reality  or  eternal  modes  of 
being.  It  would  be  necessary  to  mention  first  of  all,  as 
most  imposing  of  all,  our  modern  form  of  the  ancient 


HUMAN    SIGNIFICANCE    OF    MATHEMATICS  55 

doctrine  of  fate.  I  mean  the  reigning  conception  of  our 
universe  as  an  infinite  machine  —  a  powerful  conception 
that  more  and  more  fascinates  scientific  minds  even 
to  the  point  of  obsession  and  according  to  which  it 
should  be  possible,  were  knowledge  sufficiently  advanced, 
to  formulate,  in  a  system  of  differential  equations,  the 
whole  of  cosmic  history  from  eternity  to  eternity  in 
minutest  detail,  not  even  excluding  a  skeptic's  doubt 
whether  such  formulation  be  theoretically  possible  nor 
excluding  the  conviction,  which  some  minds  have,  that 
the  doctrine,  regarded  as  an  ultimate  creed,  is  an  abomi- 
nable libel  against  the  character  of  a  world  where  the 
felt  freedom  of  the  human  spirit  is  not  an  illusion.  It 
would  be  necessary  to  mention  —  as  next  perhaps  in 
order  of  impressiveness  —  another  doctrine  that  is,  curi- 
ously enough,  vividly  reminiscent  of  old-time  fate.  I 
allude  this  time  to  the  doctrine  of  heredity,  a  tremendous 
conception,  in  accordance  with  which  —  as  Professor  W. 
B.  Smith  has  said  in  his  recent  powerful  address  on 
"Push  or  Pull"?  —  "the  remotest  past  reaches  out  its 
skeletal  fingers  and  grapples  both  present  and  future  in 
its  iron  grip."  And  there  is  the  conservation  of  energy 
and  that  of  mass  —  both  of  them,  again,  doctrines  pre- 
figured in  the  thought  of  ancient  Greece  —  and  numer- 
ous other  so-called  natural  laws,  simple  and  complex, 
familiar  and  unfamiliar,  all  posing  as  permanent  forms 
of  reality  —  as  natural  invariants  under  the  infinite 
system  of  cosmic  transformations  —  and  thus  together 
constituting  the  enlarging  contribution  of  natural  science 
towards  the  slow  vindication  of  a  world  that  has  seemed 
capricious,  lawless  and  impermanent. 

Such,  then,  is  a  conspectus,  suggested  rather  than 
portrayed,  of  the  results  which  the  great  allies  of  mathe- 
matics, operating  through  the  ages,  have  achieved  in 


56  HUMAN    SIGNIFICANCE    OF    MATHEMATICS 

their  passionate  endeavor  to  transcend  the  tragic  vicissi- 
tudes and  limitations  of  life  in  an  "  ever-growing  and 
perishing"  universe  and  to  win  at  length  the  freedom, 
the  dignity  and  the  peace  of  a  stable  world  where  order 
and  harmony  reign  and  spiritual  goods  endure.  If  we 
are  to  arrive  at  a  really  just  or  worthy  sense  of  the 
human  significance  of  mathematics,  it  is  in  relation  with 
those  great  results  of  her  sister  enterprises  that  the 
achievements  of  this  science  must  be  appraised.  Im- 
mense indeed  and  high  is  the  task  of  criticism  as  thus 
conceived.  How  diverse  and  manifold  the  doctrines  to 
be  evaluated,  what  depths  to  be  plumbed,  what  heights 
to  be  scaled,  how  various  the  relationships  and  digni- 
ties to  be  assigned  their  rightful  place  in  the  hierarchy 
of  values.  In  the  presence  of  such  a  task  what  can  we 
think  or  say  in  the  remaining  moments  of  the  hour? 
If  we  have  succeeded  in  setting  the  problem  in  its 
proper  light  and  in  indicating  the  sole  eminence  from 
which  the  matter  may  be  rightly  viewed,  we  ought  per- 
haps to  be  content  with  that  as  the  issue  of  the  hour, 
for  it  is  worth  while  to  sketch  a  worthy  program  of 
criticism  even  if  time  fails  us  to  perform  fully  the  task 
thus  set.  And  yet  I  can  not  refrain  from  inviting  you  to 
imagine,  before  we  close,  a  few  at  least  of  the  things  that 
one  who  essayed  the  great  critique  would  submit  to  his 
auditors  for  meditation.  And  what  do  you  imagine  the 
guiding  lines  and  major  themes  of  his  discourse  would  be? 
I  fancy  he  would  say:  The  question  before  us,  ladies 
and  gentlemen,  is  not  a  question  of  weighing  utilities  nor 
of  counting  applications  nor  of  measuring  material  gains; 
it  is  a  question  of  human  ideals  together  with  the  vari- 
ous means  of  pursuing  them  and  the  differing  degrees 
of  their  approximation;  we  are  occupied  with  a  ques- 
tion of  appreciation,  with  the  problem  of  values.  I  am, 


HUMAN    SIGNIFICANCE    OF    MATHEMATICS  57 

he  would  say,  addressing  you  as  representatives  of  man, 
and  in  so  doing,  I  am  not  regarding  man  as  a  mere  prac- 
tician, as  a  hewer  of  wood  and  drawer  of  water,  as  an 
animal  content  to  serve  the  instincts  for  shelter  and 
food  and  reproduction.  I  am  contemplating  him  as  a 
spiritual  being,  as  a  thinker,  poet,  dreamer,  as  a  lover 
of  knowledge  and  beauty  and  wisdom  and  the  joy  of 
harmony  and  light,  responding  to  the  lure  of  an  ideal 
destiny,  troubled  by  the  mystery  of  a  baffling  world, 
conscious  subject  of  tragedy,  yearning  for  stable  reality, 
for  infinite  freedom,  for  perpetuity  and  a  thousand  per- 
fections of  life.  As  representatives  of  such  a  being,  you, 
he  would  say,  and  I,  even  if  we  be  not  ourselves  pro- 
ducers of  theology  or  philosophy  or  science  or  jurispru- 
dence or  art  or  mathematics,  are  nevertheless  rightful 
inheritors  of  all  this  manifold  wisdom  of  man.  The 
question  is:  What  is  the  inheritance  worth?  We  are 
the  heirs  and  we  are  to  be  the  judges  of  the  great 
responses  that  time  has  made  to  the  spiritual  needs  of 
humanity.  What  are  the  responses  worth?  What  are 
their  values,  joint  and  several,  absolute  and  relative? 
And  what,  especially,  is  the  human  worth  of  the  re- 
sponse of  mathematics?  It  is,  he  would  say,  not  only 
our  privilege,  but,  as  educated  individuals  and  especially 
as  representatives  of  our  race,  it  is  our  duty,  to  ponder 
the  matter  and  reach,  if  we  can,  a  right  appraisement. 
For  the  proper  study  of  mankind  is  man,  and  it  is 
essential  to  remember  that  ''La  vie  de  la  science  cst  la 
critique"  I  have,  he  would  say,  tried  to  make  it  clear 
that  mathematics  is  not  an  isolated  science.  I  have 
tried  to  show  that  it  is  not  an  antagonist,  nor  a  rival, 
but  is  the  comrade  and  ally  of  the  other  great  forms 
of  spiritual  activity,  all  aiming  at  the  same  high  end. 
I  have  reminded  you  of  the  principal  answers  made  by 


58  HUMAN    SIGNIFICANCE    OF   MATHEMATICS 

these  to  the  spiritual  needs  of  man,  and  I  do  not,  he 
would  say,  desire  to  underrate  or  belittle  them.  They 
are  a  precious  inheritance.  Many  of  them  have  not, 
indeed,  stood  the  test  of  time;  others  will  doubtless 
endure  for  aye;  all  of  them,  for  a  longer  or  shorter 
period,  have  softened  the  ways  of  life  to  millions  of 
men  and  women.  Neither  do  I  desire,  he  would  say, 
to  exaggerate  the  contributions  of  mathematics  to  the 
spiritual  weal  of  humanity.  What  I  desire  is  a  fair 
comparative  estimate  of  its  claims.  "Truth  is  the  be- 
ginning of  every  good  thing,  both  to  gods  and  men." 
I  am  asking  you  to  compare,  consider  and  judge  for 
yourselves.  The  task  is  arduous  and  long. 

There  are,  our  critic  would  say,  certain  paramount 
considerations  that  every  one  in  such  an  enterprise  must 
weigh,  and  a  few  of  them  may,  in  the  moments  that 
remain,  be  passed  in  brief  review.  Consider,  for  ex- 
ample, our  human  craving  for  a  world  of  stable  reality. 
Where  is  it  to  be  found?  We  know  the  answers  of  the- 
ology, of  philosophy,  of  natural  science  and  the  rest. 
We  know,  too,  the  answer  of  literature  and  general 
thought: 

The  cloud-capped  towers,  the  gorgeous  palaces, 
The  solemn  temples,  the  great  globe  itself, 
Yea,  all  which  it  inherit,  shall  dissolve, 
And,  like  the  baseless  fabric  of  this  vision, 
Leave  not  a  rack  behind. 

And  now  what,  he  would  ask,  is  the  answer  of  mathe- 
matics? The  answer,  he  would  have  to  say,  is  this: 
Transcending  the  flux  of  the  sensuous  universe,  there  exists 
a  stable  world  of  pure  thought,  a  divinely  ordered  world  of 
ideas,  accessible  to  man,  free  from  the  mad  dance  of  time, 
infinite  and  eternal. 

Consider  our  human  craving  for  freedom.  Of  free- 
dom there  are  many  kinds.  Is  it  the  freedom  of  limitless 


HUMAN    SIGNIFICANCE    OF    MATHEMATICS  59 

room,  where  our  passion  for  outward  expression,  for 
externalization  of  thought,  may  attain  its  aim?  It  is 
to  mathematics,  our  critic  would  say,  that  man  is  in- 
debted for  that  priceless  boon;  for  it  is  the  cunning  of 
this  science  that  has  at  length  contrived  to  release  our 
long  imprisoned  thought  from  the  old  confines  of  our 
three-fold  world  of  sense  and  opened  to  its  wing  the 
interminable  skies  of  hyperspace.  But  if  it  be  a  more 
fundamental  freedom  that  is  meant,  if  it  be  freedom  of 
thought  proper  —  freedom,  that  is,  for  the  creative 
activity  of  intellect  —  then  again  it  is  to  mathematics 
that  our  faculties  must  look  for  the  definition  and  a  right 
estimate  of  their  prerogatives  and  power.  For,  regard- 
ing this  matter,  we  may  indeed  acquire  elsewhere  a 
suspicion  or  an  inkling  of  the  truth,  but  mathematics, 
and  nothing  else,  is  qualified  to  give  us  knowledge  of  the 
fact  that  our  intellectual  freedom  is  absolute  save  for  a 
single  limitation  —  the  law  of  non-contradiction,  the 
law  of  logical  compatibility,  the  law  of  intellectual  har- 
mony—  sole  restriction  imposed  by  "the  nature  of 
things"  or  by  logic  or  by  the  muses  upon  the  creative 
activity  of  the  human  spirit. 

Consider  next,  the  critic  might  say,  our  human  craving 
for  a  living  sense  of  rapport  and  comradeship  with  a 
divine  Being  infinite  and  eternal.  Except  through  the 
modern  mathematical  doctrine  of  infinity,  there  is,  he 
would  have  to  say,  no  rational  way  by  which  we  may 
even  approximate  an  understanding  of  the  supernal 
attributes  with  which  our  faculty  of  idealization  has 
clothed  Deity  —  no  way,  except  this,  by  which  our 
human  reason  may  gaze  understandingly  upon  the 
downward-looking  aspects  of  the  overworld.  But  this 
is  not  all.  I  need  not,  he  would  say,  remind  you  of  the 
reverent  saying  attributed  to  Plato  that  "God  is  a 


60  HUMAN    SIGNIFICANCE    OF    MATHEMATICS 

geometrician."  Who  is  so  unfortunate  as  not  to  know 
something  of  the  religious  awe,  the  solace  and  the  peace 
that  come  from  cloistral  contemplation  of  the  purity  and 
everlastingness  of  mathematical  truth? 

Mighty  is  the  charm  of  those  abstractions  to  a  mind  beset  with  images 
and  haunted  by  himself. 

"More  frequently,"  says  Wordsworth,  speaking  of 
geometry, 

More  frequently  from  the  same  source  I  drew 

A  pleasure  quiet  and  profound,  a  sense 

Of  permanent  and  universal  sway, 

And  paramount  belief;  there,  recognized 

A  type,  for  finite  natures,  of  the  one 

Supreme  Existence,  the  surpassing  life 

Which  to  the  boundaries  of  space  and  time, 

Of  melancholy  space  and  doleful  time, 

Superior  and  incapable  of  change, 

Not  touched  by  welterings  of  passion  —  is, 

And  hath  the  name  of  God.    Transcendent  peace 

And  silence  did  wait  upon  those  thoughts 

That  were  a  frequent  comfort  to  my  youth. 

And  so  our  spokesman,  did  time  allow,  might  con- 
tinue, inviting  his  auditors  to  consider  the  relations  of 
mathematics  to  yet  other  great  ideals  of  humanity  — 
our  human  craving  for  rectitude  of  thought,  for  ideal 
justice,  for  dominion  over  the  energies  and  ways  of  the 
material  universe,  for  imperishable  beauty,  for  the  dig- 
nity and  peace  of  intellectual  harmony.  We  know  that 
in  all  such  cases  the  issue  of  the  great  critique  would 
be  the  same,  and  it  is  needless  to  pursue  the  matter 
further.  The  light  is  clear  enough.  Mathematics  is,  in 
many  ways,  the  most  precious  response  that  the  human 
spirit  has  made  to  the  call  of  the  infinite  and  eternal. 
It  is  man's  best  revelation  of  the  "Deep  Base  of  the 
World." 


THE  HUMANIZATION  OF  THE  TEACHING 
OF  MATHEMATICS1 

WHEN  the  distinguished  chairman  of  your  mathe- 
matical conference  did  me  the  honor  to  request  me  to 
speak  to  you,  he  was  generous  enough,  whether  wisely 
or  unwisely,  to  leave  the  choice  of  a  subject  to  my  dis- 
cretion, merely  stipulating  that,  whatever  the  title  might 
be,  the  address  itself  should  bear  upon  the  professional 
function  of  those  men  and  women  who  are  engaged 
in  teaching  mathematics  in  secondary  schools.  Inex- 
pertness,  it  has  been  said,  is  the  curse  of  the  world; 
and  one  may,  not  unnaturally,  feel  some  hesitance  in 
undertaking  a  task  that  might  seem  to  resemble  the 
rftle  of  a  physician  when,  as  sometimes  happens,  he  is 
called  upon  to  treat  a  patient  whose  health  and  medical 
competence  surpass  his  own.  I  trust  I  am  not  wanting 
in  that  natural  feeling.  In  the  present  instance  two 
considerations  have  enabled  me  to  overcome  it.  One 
of  them  is  that,  having  had  some  experience  in  teaching 
mathematics  in  secondary  schools,  I  might,  it  seemed 
to  me,  regard  that  experience,  though  it  was  gained 
more  than  a  score  of  years  ago,  as  giving  something 
like  a  title  to  be  heard  in  your  counsels.  The  other 
consideration  is  that,  in  regard  to  the  teaching  of  mathe- 
matics, whether  in  secondary  schools  or  in  colleges,  I 

1  Address  given  at  the  meeting  of  the  Michigan  School  Masters'  Club, 
at  Ann  Arbor,  March  28,  1912.  Printed  in  Science,  April  26,  1912;  in 
The  Educational  Renew,  September,  1912;  and  in  the  Michigan  School 
Masters'  Magazine. 


62  HUMANIZATION  OF   TEACHING  MATHEMATICS 

have  acquired  a  certain  conviction,  a  pretty  firm  con- 
viction, which,  were  it  properly  presented,  you  would 
doubtless  be  generous  enough  and  perhaps  ingenious 
enough  to  regard  as  having  some  sort  of  likeness  to  a 
message. 

My  conviction  is,  that  hope  of  improvement  in  mathe- 
matics teaching,  whether  in  secondary  schools  or  in 
colleges,  lies  mainly  in  the  possibility  of  humanizing  it. 
It  is  worth  while  to  remember  that  our  pupils  are  hu- 
man beings.  What  it  means  to  be  a  human  being  we 
all  of  us  presumably  know  pretty  well;  indeed  we  know 
it  so  well  that  we  are  unable  to  tell  it  to  one  another 
adequately;  and,  just  because  we  do  so  well  know  what 
it  means  to  be  a  human  being,  we  are  prone  to  forget 
it  as  we  forget,  except  when  the  wind  is  blowing,  that 
we  are  constantly  immersed  in  the  earth's  atmosphere. 
To  humanize  the  teaching  of  mathematics  means  so  to 
present  the  subject,  so  to  interpret  its  ideas  and  doc- 
trines, that  they  shall  appeal,  not  merely  to  the  com- 
putatory  faculty  or  to  the  logical  faculty  but  to  all  the 
great  powers  and  interests  of  the  human  mind.  That 
mathematical  ideas  and  doctrines,  whether  they  be  more 
elementary  or  more  advanced,  admit  of  such  a  manifold, 
liberal  and  stimulating  interpretation,  and  that  there- 
fore the  teaching  of  mathematics,  whether  in  secondary 
schools  or  in  colleges,  may  become,  in  the  largest  and 
best  sense,  human,  I  have  no  doubt.  That  mathe- 
matical ideas  and  doctrines  do  but  seldom  receive  such 
interpretation  and  that  accordingly  the  teaching  of 
mathematics  is  but  seldom,  in  the  largest  and  best  sense, 
human,  I  believe  to  be  equally  certain.  That  the  indi- 
cated humanization  of  mathematical  teaching,  the  bring- 
ing of  the  matter  and  the  spirit  of  mathematics  to  bear, 
not  merely  upon  certain  fragmentary  faculties  of  the 


HUMANIZATION  OP  TEACHING   MATHEMATICS  63 

mind,  but  upon  the  whole  mind,  that  this  is  a  great 
desideration  is,  I  assume,  beyond  dispute. 

How  can  such  humanization  be  brought  about?  The 
answer,  I  believe,  is  not  far  to  seek.  I  do  not  mean  that 
the  answer  is  easy  to  discover  or  easy  to  communicate. 
I  mean  that  the  game  is  near  at  hand  and  that  it  is 
not  difficult  to  locate  it,  though  it  may  not  be  easy  to 
capture  it.  The  difficulty  inheres,  I  believe,  in  our  con- 
ception of  mathematics  itself;  not  so  much  in  our  con- 
ception of  what  mathematics,  in  a  definitional  sense,  is, 
for  that  sense  of  what  mathematics  is  has  become  pretty 
clear  in  our  day,  but  in  our  sense  or  want  of  sense  of 
what  mathematics,  whatever  it  may  be,  humanly  sig- 
nifies. In  order  to  humanize  mathematical  teaching  it  is 
necessary,  and  I  believe  it  is  sufficient,  to  come  under 
the  control  of  a  right  conception  of  the  human  sig- 
nificance of  mathematics.  It  is  sufficient,  I  mean  to  say, 
and  it  is  necessary,  greatly  to  enlarge,  to  enrich  and  to 
vitalize  our  sense  of  what  mathematics,  regarded  as 
human  enterprise,  signifies. 

What  does  mathematics,  regarded  as  an  enterprise  of 
the  human  spirit,  signify?  What  is  a  just  and  worthy 
sense  of  the  human  significance  of  mathematics? 

To  the  extent  in  which  any  of  us  really  succeeds  in 
answering  that  question  worthily,  his  teaching  will  have 
the  human  quality,  in  so  far  as  his  teaching  is,  in  point 
of  external  circumstance,  free  to  be  what  it  would.  I 
believe  it  is  important  to  put  the  question,  and  it  is 
with  the  putting  of  it  rather  than  with  the  proposing  of 
an  answer  to  it  that  I  am  here  at  the  outset  mainly 
concerned.  For  any  one  who  is  really  to  acquire  pos- 
session of  an  answer  that  is  worthy  must  win  the  answer 
for  himself.  I  need  not  say  to  you  that  such  an  acqui- 
sition as  a  worthy  answer  to  this  kind  of  question  does 


64         HUMANIZATION   OF   TEACHING   MATHEMATICS 

not  belong  to  the  category  of  things  that  may  be  lent 
or  borrowed,  sold  or  bought,  donated  or  acquired  by 
gift.  No  doubt  the  answers  we  may  severally  win  will 
differ  as  our  temperaments  differ.  Yet  the  matter  is 
not  solely  a  matter  of  temperament.  It  is  much  more 
a  matter  first  of  knowledge  and  then  of  the  evaluation 
of  the  knowledge  and  of  its  subject.  To  the  winning 
of  a  worthy  sense  of  the  human  significance  of  mathe- 
matics two  things  are  indispensable,  knowledge  and  re- 
flection: knowledge  of  mathematics  and  reflection  upon 
it.  To  the  winning  of  such  a  sense  it  is  essential  to 
have  the  kind  of  knowledge  that  none  but  serious 
students  of  mathematics  can  gain.  Equally  essential 
is  another  thing  and  this  thing  students  of  mathematics 
in  our  day  do  not,  or  do  but  seldom,  gain.  I  mean  the 
kind  of  insight  and  the  liberality  of  view  that  are  to  be 
acquired  only  by  prolonged  contemplation  of  the  nature 
of  mathematics  and  by  prolonged  reflection  upon  its 
relations  of  contrast  and  similitude  to  the  other  great 
forms  of  spiritual  activity. 

The  question,  though  it  is  a  question  about  mathe- 
matics, is  not  a  mathematical  question;  it  is  a  philo- 
sophical question.  And  just  because  it  is  a  philosophical 
question,  mathematicians,  despite  the  fact  that  one 
of  the  indispensable  qualifications  for  considering  it  is 
possessed  by  them  alone,  have  in  general  ignored  it. 
They  have,  in  general,  ignored  it,  and  their  ignoring  of 
it  may  help  to  explain  the  curious  paradox  that  whilst 
the  world,  whose  mathematical  knowledge  varies  from 
little  to  less,  has  always  as  if  instinctively  held  mathe- 
matical science  in  high  esteem,  it  has  at  the  same  time 
usually  regarded  mathematicians  as  eccentric  and  ab- 
normal, as  constituting  a  class  apart,  as  being  something 
more  or  something  less  than  human.  It  may  explain, 


HUMANIZATION   OF   TEACHING   MATHEMATICS  65 

too,  I  venture  to  believe  it  does  partly  explain,  both 
why  it  is  that  in  the  universities  the  number  of  students 
attracted  to  advanced  lectures  in  mathematics  compared 
with  the  numbers  drawn  to  advanced  courses  in  some 
other  great  subjects  not  inherently  more  attractive,  is 
so  small;  and  why  it  is  that,  among  the  multitudes  who 
pursue  mathematics  in  the  secondary  schools,  only  a  few 
find  in  the  subject  anything  like  delight.  For  I  do  not 
accept  the  traditional  and  still  current  explanation,  that 
the  phenomenon  is  due  to  a  well-nigh  universal  lack 
of  mathematical  faculty.  I  maintain,  on  the  contrary, 
that  a  vast  majority  of  mankind  possess  mathematical 
faculty  in  a  very  considerable  degree.  That  the  average 
pupil's  interest  in  mathematics  is  but  slight,  is  a  matter 
of  common  knowledge.  His  lack  of  interest  is,  in  my 
opinion,  due,  not  to  a  lack  of  the  appropriate  faculty 
in  him,  but  to  the  circumstance  that  he  is  a  human 
being,  whilst  mathematics,  though  it  teems  with  human 
interest,  is  not  presented  to  him  in  its  human  guise. 

If  you  ask  the  world  —  represented,  let  us  say,  by 
the  man  in  the  street  or  in  the  market  place  or  the 
field  —  to  tell  you  its  estimate  of  the  human  significance 
of  mathematics,  the  answer  of  the  world  will  be,  that 
mathematics  has  given  mankind  a  metrical  and  com- 
putatory  art  essential  to  the  effective  conduct  of  daily 
life,  that  mathematics  admits  of  countless  applications 
in  engineering  and  the  natural  sciences,  and  finally  that 
mathematics  is  a  most  excellent  instrumentality  for 
giving  mental  discipline.  Such  will  be  the  answer  of  the 
world.  The  answer  is  intelligible,  it  is  important,  and 
it  is  good  so  far  as  it  goes;  but  it  is  far  from  going  far 
enough  and  it  is  not  intelligent.  That  it  is  far  from 
going  far  enough  will  become  evident  as  we  proceed. 
That  the  answer  is  not  intelligent  is  evident  at  once, 


66  HUMANIZATION  OF   TEACHING  MATHEMATICS 

for  the  first  part  of  it  seems  to  imply  that  the  rudi- 
mentary mathematics  of  the  carpenter  and  the  counting- 
house  is  scientific,  which  it  is  not;  the  second  part  of 
the  answer  is  but  an  echo  by  the  many  of  the  voice 
of  the  few;  and,  as  to  the  final  part,  the  world's  con- 
ception of  intellectual  discipline  is  neither  profound  nor 
well  informed  but  is  itself  in  sorry  need  of  discipline. 

If,  turning  from  the  world  to  a  normal  mathematician, 
you  ask  him  to  explain  to  you  the  human  significance 
of  mathematics,  he  will  repeat  to  you  the  answer  of  the 
world,  of  course  with  far  more  appreciation  than  the 
world  has  of  what  the  answer  means,  and  he  will  sup- 
plement the  world's  response  by  an  important  addition. 
He  will  add,  that  is,  that  mathematics  is  the  exact 
science,  the  science  of  exact  thought  or  of  rigorous 
thinking.  By  this  he  will  not  mean  what  the  world 
would  mean  if  the  world  employed,  as  sometimes  it  does 
employ,  the  same  form  of  words.  He  will  mean  some- 
thing very  different.  Especially  if  he  be,  as  I  suppose 
him  to  be,  a  normal  mathematician  of  the  modern 
critical  type,  he  will  mean  that  mathematics  is,  in  the 
oft-cited  language  of  Benjamin  Peirce,  "the  science 
that  draws  necessary  conclusions;"  he  will  mean  that, 
in  the  felicitous  words  of  William  Benjamin  Smith, 
"mathematics  is  the  universal  art  apodictic;"  he  will 
mean  that  mathematics  is,  in  the  nicely  technical  phrase 
of  Fieri,  "a  hypothetico-deductive  system."  If  you  ask 
him  whether  mathematics  is  the  science  of  rigorous 
thinking  about  all  the  things  that  engage  the  thought  of 
mankind  or  only  about  a  few  of  them,  such  as  numbers, 
figures,  certain  operations,  and  the  like,  the  answer  he 
will  give  you  depends.  If  he  be  a  normal  mathematician 
of  the  elder  school,  he  will  say  that  mathematics  is  the 
science  of  rigorous  thinking  about  only  a  relatively  few 


HUMANIZATION  OF   TEACHING   MATHEMATICS  67 

things  and  that  these  are  such  as  you  have  exemplified. 
And  if  now,  with  a  little  Socratic  persistence,  you  press 
him  to  indicate  the  human  significance  of  a  science  of 
rigorous  thinking  about  only  a  few  of  the  countless 
things  that  engage  human  thought,  his  answer  will  give 
you  but  little  beyond  a  repetition  of  the  above-mentioned 
answer  of  the  world.  But  if  he  be  a  normal  mathe- 
matician of  the  modern  critical  type,  he  will  say  that 
mathematics  is  the  science  of  rigorous  thinking  about 
all  the  things  that  engage  human  thought,  about  all  of 
them,  he  will  mean,  in  the  sense  that  thinking,  as  it 
approaches  perfection,  tends  to  assume  certain  definite 
forms,  that  these  forms  are  the  same  whatever  the 
subject  matter  of  the  thinking  may  be,  and  that  mathe- 
matics is  the  science  of  these  forms  as  forms.  If  you 
respond,  as  you  well  may  respond,  that,  in  accordance 
with  this  ontological  conception  of  mathematics,  this 
science,  instead  of  thinking  about  all,  thinks  about 
none,  of  the  concrete  things  of  interest  to  human 
thought,  and  that  accordingly  Mr.  Bertrand  Russell 
was  right  in  saying  that  "mathematics  is  the  science 
in  which  one  never  knows  what  one  is  talking  about  nor 
whether  what  one  says  is  true"  -if  you  respond  that, 
from  the  point  of  view  above  assumed,  that  delicious 
mot  of  Mr.  Russell's  must  be  solemnly  held  as  true,  and 
then  if,  in  accordance  with  your  original  purpose,  you 
once  more  press  for  an  estimation  of  the  human  sig- 
nificance of  such  a  science,  I  fear  that  the  reply,  if  your 
interlocutor  is  a  mathematician  of  the  normal  type,  will 
contain  little  that  is  new  beyond  the  assertion  that  the 
science  in  question  is  very  interesting,  where,  by  in- 
teresting, he  means,  of  course,  interesting  to  mathe- 
maticians. It  is  true  that  Professor  Klein  has  said: 
"Apart  from  the  fact  that  pure  mathematics  can  not  be 


68  HUMANIZATION   OF   TEACHING   MATHEMATICS 

supplanted  by  anything  else  as  a  means  for  developing 
the  purely  logical  faculties  of  the  mind,  there  must  be 
considered  here  as  elsewhere  the  necessity  of  the  pres- 
ence of  a  few  individuals  in  each  country  developed  in 
a  far  higher  degree  than  the  rest,  for  the  purpose  of 
keeping  up  and  gradually  raising  the  general  standard. 
Even  a  slight  raising  of  the  general  level  can  be  ac- 
complished only  when  some  few  minds  have  progressed 
far  ahead  of  the  average."  Here  indeed  we  have,  in 
these  words  of  Professor  Klein,  a  hint,  if  only  a  hint,  of 
something  better.  But  Professor  Klein  is  not  a  mathe- 
matician of  the  normal  type,  he  is  hypernormal.  If,  in 
order  to  indicate  the  human  significance  of  mathematics 
regarded  as  the  science  of  the  forms  of  thought  as  forms, 
your  normal  mathematician  were  to  say  that  these  forms 
constitute,  of  themselves,  an  infinite  and  everlasting 
world  whose  beauty,  though  it  is  austere  and  cold,  is 
pure,  and  in  which  is  the  secret  and  citadel  of  whatever 
order  and  harmony  our  concrete  universe  contains,  it 
would  yet  be  your  right  and  your  duty  to  ask,  as  the 
brilliant  author  of  "East  London  Visions"  once  asked 
me,  namely,  what  is  the  human  significance  of  "this 
majestic  intellectual  cosmos  of  yours,  towering  up  like 
a  million-lustered  iceberg  into  the  arctic  night,"  seeing 
that,  among  mankind,  none  is  permitted  to  behold  its 
more  resplendent  wonders  save  the  mathematician  him- 
self? But  the  normal  mathematician  will  not  say  what 
I  have  just  now  supposed  him  to  say ;  he  will  not  say  it, 
because  he  is,  by  hypothesis,  a  normal  mathematician, 
and  because,  being  a  normal  mathematician,  he  is  exclu- 
sively engaged  in  exploring  the  iceberg.  A  farmer  was 
once  asked  why  he  raised  so  many  hogs.  "In  order," 
he  said,  "to  buy  more  land."  Asked  why  he  desired 
more  land,  his  answer  was,  "in  order  to  raise  more 


HUMANIZATION   OF   TEACHING   MATHEMATICS  69 

corn."  Being  asked  to  say  why  he  would  raise  more 
corn,  he  replied  that  he  wished  to  raise  more  hogs.  If 
you  ask  the  normal  mathematician  why  he  explores  the 
iceberg  so  much,  his  answer  will  be,  in  effect  at  least, 
"in  order  to  explore  it  more."  In  this  exquisite  cir- 
cularity of  motive,  the  farmer  and  the  normal  mathe- 
matician are  well  within  their  rights.  They  are  within 
their  rights  just  as  a  musician  would  be  within  his 
rights  if  he  chanced  to  be  so  exclusively  interested  in 
the  work  of  composition  as  never  to  be  concerned 
with  having  his  creations  rendered  before  the  public 
and  never  to  attempt  a  philosophic  estimate  of  the 
human  worth  of  music.  The  distinction  involved  is 
not  the  distinction  between  human  and  inhuman, 
between  social  and  anti-social;  it  is  the  distinction 
between  what  is  human  or  inhuman,  social  or  anti- 
social, and  what  is  neither  the  one  nor  the  other. 
No  one,  I  believe,  may  contest  the  normal  mathema- 
tician's right  as  a  mathematical  student  or  investigator 
to  be  quite  indifferent  as  to  the  social  value  or  the 
human  worth  of  his  activity.  Such  activity  is  to 
be  prized  just  as  we  prize  any  other  natural  agency 
or  force  that,  however  undesignedly,  yet  contributes, 
sooner  or  later,  directly  or  indirectly,  to  the  weal  of 
mankind.  The  fact  is  that,  among  motives  in  research, 
scientific  curiosity,  which  is  neither  moral  nor  immoral, 
is  far  more  common  and  far  more  potent  than  charity 
or  philanthropy  or  benevolence.  But  when  the  mathe- 
matician passes  from  the  rdle  of  student  or  investigator 
to  the  role  of  teacher,  that  right  of  indifference  ceases, 
for  he  has  passed  to  an  office  whose  functions  are  social 
and  whose  obligations  are  human.  It  is  not  his  privi- 
lege to  chill  and  depress  with  the  encasing  fogs  of  the 
iceberg.  It  is  his  privilege  and  his  duty,  in  so  far  as 


7O  HUMANIZATION   OF   TEACHING   MATHEMATICS 

he  may,  to  disclose  its  "  million-lustered "  splendors  in 
all  their  power  to  quicken  and  illuminate,  to  charm  and 
edify,  the  whole  mind. 

The  conception  of  mathematics  as  the  science  of  the 
forms  of  thought  as  forms,  the  conception  of  it  as  the 
refinement,  prolongation  and  elaboration  of  pure  logic, 
is,  as  you  are  doubtless  aware,  one  of  the  great  out- 
comes, perhaps  I  should  say  it  is  the  culminating  philo- 
sophical outcome,  of  a  century's  effort  to  ascertain  what 
mathematics,  in  its  intimate  structure,  is.  This  concep- 
tion of  what  mathematics  is  comes  to  its  fullest  expres- 
sion and  best  defense,  as  you  doubtless  know,  in  such 
works  as  Schroeder's  "Algebra  der  Logik,"  White- 
head's  " Universal  Algebra,"  Russell's  "Principles  of 
Mathematics,"  Peano's  "Formulario  Matematico,"  and 
especially  in  Whitehead  and  Russell's  monumental 
"Principia  Mathematical'  I  cite  this  literature  because 
it  tells  us  what,  in  a  definitional  sense,  the  science  in 
which  the  normal  mathematician  is  exclusively  engaged, 
is.  If  we  wish  to  be  told  what  that  science  humanly 
signifies,  we  must  look  elsewhere;  we  must  look  to  a 
mathematician  like  Plato,  for  example,  or  to  a  phi- 
losopher like  Poincare,  but  especially  must  we  look  to 
our  own  faculty  for  discerning  those  fine  connective 
things  —  community  of  aim,  interformal  analogies,  struc- 
tural similitudes  —  that  bind  all  the  great  forms  of 
human  activity  and  aspiration  —  natural  science,  the- 
ology, philosophy,  jurisprudence,  religion,  art  and  mathe- 
matics —  into  one  grand  enterprise  of  the  human  spirit. 

In  the  autumn  of  1906  there  was  published  in  Poet 
Lore  a  short  poem  which,  though  it  says  nothing  ex- 
plicitly of  mathematics,  yet  admits  of  an  interpretation 
throwing  much  light  upon  the  human  significance  of  the 
science  and  indicating  well,  I  think,  the  normal  mathe- 


HUlfANIZATION   OF  TEACHING   MATHEMATICS  71 

matician's  place  in  the  world  of  spiritual  interests.  Jlhe 
author  of  the  poem  is  my  excellent  friend  and  teacher, 
Professor  William  Benjamin  Smith,  mathematician,  phi- 
losopher, poet  and  theologian.  I  have  not  asked  his 
permission  to  interpret  the  poem  as  I  shall  invite  you 
to  interpret  it.  What  its  original  motive  was  I  am  not 
informed  —  it  may  have  been  the  exceeding  beauty  of 
the  ideas  expressed  in  it  or  the  harmonious  mingling 
of  their  light  with  the  melody  of  their  song.  The  title 
of  the  poem  is  "The  Merman  and  the  Seraph."  As 
you  listen  to  the  reading  of  it,  I  shall  ask  you  to  regard 
the  Merman  as  representing  the  normal  mathematician 
and  the  Seraph  as  representing,  let  us  say,  the  life  of 
the  emotions  in  their  higher  reaches  and  their  finer 
susceptibilities. 

Deep  the  sunless  seas  amid, 
Far  from  Man,  from  Angel  hid, 
Where  the  soundless  tides  are  rolled 
Over  Ocean's  treasure-hold, 
With  dragon  eye  and  heart  of  stone, 
The  ancient  Merman  mused  alone. 

And  aye  his  arrowed  Thought  he  wings 

Straight  at  the  inmost  core  of  things  — 

As  mirrored  in  his  Magic  glass 

The  lightning-footed  Ages  pass,  — 

And  knows  nor  joy  nor  Earth's  distress, 

But  broods  on  Everlastingness. 

"Thoughts  that  love  not,  thoughts  that  hate  not, 

Thoughts  that  Age  and  Change  await  not, 

All  unfeeling, 

All  revealing, 

Scorning  height's  and  depth's  concealing, 
These  be  mine  —  and  these  alone  I "  — 
Saith  the  Merman's  heart  of  stone. 

Flashed  a  radiance  far  and  nigh 
As  from  the  vertex  of  the  sky,  — 
Lo!  a  Maiden  beauty-bright 


72  HUMANIZATION   OF   TEACHING   MATHEMATICS 

And  mantled  with  mysterious  might 
Of  every  power,  below,  above, 
That  weaves  resistless  spell  of  Love. 

Through  the  weltering  waters  cold 
Shot  the  sheen  of  silken  gold; 
Quick  the  frozen  Heart  below 
Kindled  in  the  amber  glow; 
Trembling  Heavenward  Nekkan  yearned 
Rose  to  where  the  Glory  burned. 
"Deeper,  bluer  than  the  skies  are, 
Dreaming  meres  of  morn  thine  eyes  are 

All  that  brightens 

Smile  or  heightens 

Charm  is  thine,  all  life  enlightens, 
Thou  art  all  the  soul's  desire."  — 
Sang  the  Merman's  Heart  of  Fire. 
"Woe  thee,  Nekkan!    Ne'er  was  given 
Thee  to  walk  the  ways  of  Heaven; 

Vain  the  vision, 

Fate's  derision, 

Thee  that  raps  to  realms  elysian, 
Fathomless  profounds  are  thine"  — 
Quired  the  answering  voice  divine. 

Came  an  echo  from  the  West, 
Pierced  the  deep  celestial  breast; 
Summoned,  far  the  Seraph  fled, 
Trailing  splendors  overhead; 
Broad  beneath  her  flying  feet 
Laughed  the  silvered  ocean-street. 

On  the  Merman's  mortal  sight 

Instant  fell  the  pall  of  Night; 

Sunk  to  the  sea's  profoundest  floor 

He  dreams  the  vanished  Vision  o'er, 

Hears  anew  the  starry  chime, 

Ponders  aye  Eternal  Time. 

"Thoughts  that  hope  not,  thoughts  that  fear  not, 

Thoughts  that  Man  and  Demon  veer  not 

Times  unending 

Comprehending, 

Space  and  worlds  of  worlds  transcending, 
These  are  mine  —  but  these  alone!"  — 
Sighs  the  Merman's  heart  of  stone. 


HT7KANIZATION  OF   TEACHING   MATHEMATICS  73 

I  have  said  that  the  poem,  if  it  receive  the  interpre- 
tation that  I  have  invited  you  to  give  it,  throws  much 
light  on  the  human  significance  of  mathematics  and 
indicates  well  the  place  of  the  normal  mathematician 
in  the  world  of  spiritual  interests.  No  doubt  the 
place  of  the  merman  and  the  place  of  the  angel 
are  not  the  same:  no  doubt  the  world  of  whatsoever 
in  thought  is  passionless,  infinite  and  everlasting,  and 
the  world  of  whatsoever  in  feeling  is  high  and  beau- 
teous and  good  are  distinct  worlds,  and  they  are 
sundered  wide  in  the  poem.  But,  though  in  the 
poem  they  are  held  widely  apart,  in  the  poet  they  are 
united.  For  the  song  is  not  the  merman's  song  nor 
are  its  words  the  words  of  the  seraph.  It  is  the  voice 
of  the  poet  —  a  voice  of  man.  The  merman's  world 
and  the  world  of  the  seraph  are  not  the  same,  they  are 
very  distinct;  in  conception  they  are  sundered;  they 
may  be  sundered  in  life,  but  in  life  it  need  not  be  so. 
The  merman  indeed  is  confined  to  the  one  world  and 
the  seraph  to  the  other,  but  man,  a  man  unless  he  be 
a  merman,  may  inhabit  them  both.  For  the  angel's 
denial,  the  derision  of  fate,  is  not  spoken  of  man,  it  is 
spoken  of  the  merman;  and  the  merman's  sigh  is  not 
his  own,  it  is  a  human  sigh  —  so  lonely  seems  the  mer- 
man in  the  depths  of  his  abode. 

No,  the  world  of  interests  of  the  human  spirit  is  not 
the  merman's  world  alone  nor  the  seraph's  alone.  It  is 
not  so  simple.  It  is  rather  a  cluster  of  worlds,  of  worlds 
that  differ  among  themselves  as  differ  the  lights  by 
which  they  are  characterized.  As  differ  the  lights. 
The  human  spirit  is  susceptible  of  a  variety  of  lights 
and  it  lives  at  once  in  a  corresponding  variety  of  worlds. 
There  is  perception's  light,  commonly  identified  with 
solar  radiance  or  with  the  radiance  of  sound,  for  music, 


74  HUMANIZATION   OF   TEACHING  MATHEMATICS 

too,  is,  to  the  spirit,  a  kind  of  illumination:  percep- 
tional light,  in  which  we  behold  the  colors,  forms  and 
harmonies  of  external  nature:  a  beautiful  revelation  — 
a  world  in  which  any  one  might  be  willing  to  spend  the 
remainder  of  his  days  if  he  were  but  permitted  to  live 
so  long.  And  there  is  imagination's  light,  disclosing  a 
new  world  filled  with  wondrous  things,  things  that  may 
or  may  not  resemble  the  things  revealed  in  perception's 
light  but  are  never  identical  with  them:  light  that  is 
not  superficial  nor  constrained  to  paths  that  are  straight 
but  reveals  the  interiors  of  what  it  illuminates  and 
phases  that  look  away.  Again,  there  is  the  light  of 
thought,  of  reason,  of  logic,  the  light  of  analysis,  far 
dimmer  than  perception's  light,  dimmer,  too,  than  that 
of  imagination,  but  far  more  penetrating  and  far  more 
ubiquitous  than  either  of  them,  disclosing  things  that 
curiously  match  the  things  that  they  disclose  and  count- 
less things  besides,  namely,  the  world  of  ideas  and  the 
relations  that  bind  them:  a  cosmic  world,  in  the  center 
whereof  is  the  home  of  the  merman.  There  remains  to 
be  named  a  fourth  kind  of  light.  I  mean  the  light  of 
emotion,  the  radiance  and  glory  of  things  that,  save  by 
gleams  and  intimations,  are  not  revealed  in  perception 
or  in  imagination  or  in  thought:  the  light  of  the  seraph's 
world,  the  world  of  the  good,  the  true  and  the  beautiful, 
of  the  spirit  of  art,  of  aspiration  and  of  religion. 

Such,  in  brief,  is  the  cluster  of  worlds  wherein  dwell 
the  spiritual  interests  of  the  human  beings  to  whom 
it  is  our  mission  to  teach  mathematics.  My  thesis  is 
that  it  is  our  privilege  to  show,  in  the  way  of  our  teach- 
ing it,  that  its  human  significance  is  not  confined  to 
one  of  the  worlds  but,  like  a  subtle  and  ubiquitous 
ether,  penetrates  them  all.  Objectively  viewed,  con- 
ceptually taken,  these  worlds,  unlike  the  spheres  of  the 


HUMANIZAT1ON   OP  TEACHING  MATHEMATICS  75 

geometrician,  do  not  intersect  —  a  thing  in  one  of 
them  is  not  in  another;  but  the  things  in  one  of  them 
and  the  things  in  another  may  own  a  fine  resemblance 
serving  for  mutual  recall  and  illustration,  effecting 
transfer  of  attention  —  transformation  as  the  mathe- 
maticians call  it  —  from  world  to  world;  for  whilst  these 
worlds  of  interest,  objectively  viewed,  have  naught  in 
common,  yet  subjectively  they  are  united,  united  as 
differing  mansions  of  the  house  of  the  human  spirit. 
A  relation,  for  example,  between  three  independent 
variables  exists  only  in  the  grey  light  of  thought,  only 
in  the  world  of  the  merman;  the  habitation  of  the 
geometric  locus  of  the  relation  is  the  world  of  imagina- 
tion; if  a  model  of  the  locus  be  made  or  a  drawing  of 
it,  this  will  be  a  thing  in  the  world  of  perception; 
finally,  the  wondrous  correlation  of  the  three  things,  or 
the  spiritual  qualities  of  them  —  the  sensuous  beauty 
of  the  model  or  the  drawing,  the  unfailing  validity  of 
the  given  relation  holding  as  it  does  throughout  "the 
cycle  of  the  eternal  year,"  the  immobile  presence  of  the 
locus  or  image  poised  there  in  eternal  calm  like  a  figure 
of  justice  —  these  may  serve,  in  contemplating  them,  to 
evoke  the  radiance  of  the  seraph's  world:  and  thus  the 
circuit  and  interplay,  ranging  through  the  world  of 
imagination  and  the  world  of  thought  from  what  is 
sensuous  to  what  is  supernal,  is  complete.  It  would  not 
have  seemed  to  Plato,  as  it  may  seem  to  us,  a  far  cry 
from  the  prayer  of  a  poet  to  the  theorem  of  Pythagoras, 
for  example,  or  to  that  of  Archimedes  respecting  a  sphere 
and  its  circumscribing  cylinder.  Yet  I  venture  to  say, 
that  calm  reflection  upon  the  existence  and  nature  of 
such  a  theorem  —  cloistral  contemplation,  I  mean,  of  the 
fact  that  it  is  really  true,  of  its  serene  beauty,  of 
its  silent  omnipresence  throughout  the  infinite  universe 


76  HUMANIZATION   OF   TEACHING   MATHEMATICS 

of  space,  of  the  absolute  exactitude  and  invariance  of 
its  truth  from  everlasting  to  everlasting  —  such  reflec- 
tion will  not  fail  to  yield  a  sense  of  reverence  and  awe 
akin  to  the  feeling  that,  for  example,  pervades  this 
choral  prayer  by  Sophocles: 

"Oh!  that  my  lot  may  lead  me  in  the  path  of  holy 
innocence  of  word  and  deed,  the  path  which  august 
laws  ordain,  laws  that  in  the  highest  empyrean  had 
their  birth,  of  which  Heaven  is  the  father  alone,  nor 
did  the  race  of  mortal  men  beget  them,  nor  shall  oblivion 
put  them  to  sleep.  The  god  is  mighty  in  them  and  he 
groweth  not  old." 

But  why  should  we  think  it  strange  that  interests, 
though  they  seem  to  cluster  about  opposite  poles,  are 
yet  united  by  a  common  mood?  Of  the  great  world  of 
human  interests,  mathematics  is  indeed  but  a  part; 
but  is  a  central  part,  and,  in  a  profound  and  precious 
sense,  it  is  "the  eternal  type  of  the  wondrous  whole." 
For  poetry  and  painting,  sculpture  and  music  —  art 
in  all  its  forms  —  philosophy,  theology,  religion  and 
science,  too,  however  passional  their  life  and  however 
tinged  or  deeply  stained  by  local  or  temporal  circum- 
stance, yet  have  this  in  common:  they  all  of  them  aim 
at  values  which  transcend  the  accidents  and  limitations 
of  every  time  and  place;  and  so  it  is  that  the  passion- 
lessness  of  the  merman's  thought,  the  infiniteness  of 
the  kind  of  being  he  contemplates  and  the  everlasting- 
ness  of  his  achievements  enter  as  essential  qualities 
into  the  ideals  that  make  the  glory  of  the  seraph's 
world.  I  do  not  forget,  in  saying  this,  that,  of  all 
theory,  mathematical  theory  is  the  most  abstract. 
I  do  not  forget  that  mathematics  therefore  lends 
especial  sharpness  to  the  contrast  in  the  Mephistophe- 
lian  warning: 


HUMAN1ZAT10N   OF   TEACHING   MATHEMATICS  ^^ 

Grey,  my  dear  friend,  is  all  theory, 
Green  the  golden  tree  of  life. 

Yet  I  know  that  one  who  loves  not  the  grey  of  a 
naked  woodland  has  much  to  learn  of  the  esthetic  re- 
sources of  our  northern  clime.  A  mathematical  doctrine, 
taken  in  its  purity,  is  indeed  grey.  Yet  such  a  doctrine, 
a  world-filling  theory  woven  of  grey  relationships  finer 
than  gossamer  but  stronger  than  cables  of  steel,  leaves 
upon  an  intersecting  plane  a  tracery  surpassing  in  fine- 
ness and  beauty  the  exquisite  artistry  of  frost-work  upon 
a  windowpane.  Architecture,  it  has  been  said,  is  frozen 
music.  Be  it  so.  Geometry  is  frozen  architecture. 

No,  the  belief  that  mathematics,  because  it  is  abstract, 
because  it  is  static  and  cold  and  grey,  is  detached  from 
life,  is  a  mistaken  belief.  Mathematics,  even  in  its 
purest  and  most  abstract  estate,  is  not  detached  from 
life.  It  is  just  the  ideal  handling  of  the  problems  of 
life,  as  sculpture  may  idealize  a  human  figure  or  as 
poetry  or  painting  may  idealize  a  figure  or  a  scene. 
Mathematics  is  precisely  the  ideal  handling  of  the  prob- 
lems of  life,  and  the  central  ideas  of  the  science,  the 
great  concepts  about  which  its  stately  doctrines  have 
been  built  up,  are  precisely  the  chief  ideas  with  which 
life  must  always  deal  and  which,  as  it  tumbles  and 
rolls  about  them  through  time  and  space,  give  it  its 
interests  and  problems,  and  its  order  and  rationality. 
That  such  is  the  case  a  few  indications  will  suffice  to 
show.  The  mathematical  concepts  of  constant  and 
variable  are  represented  familiarly  in  life  by  the  notions 
of  fixedness  and  change.  The  concept  of  equation  or 
that  of  an  equational  system,  imposing  restriction  upon 
variability,  is  matched  in  life  by  the  concept  of  natural 
and  spiritual  law,  giving  order  to  what  were  else  chaotic 


78  HUMANIZATION  OF   TEACHING  MATHEMATICS 

change  and  providing  partial  freedom  in  lieu  of  none 
at  all.  What  is  known  in  mathematics  under  the  name 
of  limit  is  everywhere  present  in  life  in  the  guise  of 
some  ideal,  some  excellence  high-dwelling  among  the 
rocks,  an  "ever  flying  perfect"  as  Emerson  calls  it, 
unto  which  we  may  approximate  nearer  and  nearer, 
but  which  we  can  never  quite  attain,  save  in  aspiration. 
The  supreme  concept  of  functionality  finds  its  correlate 
in  life  in  the  all-pervasive  sense  of  interdependence 
and  mutual  determination  among  the  elements  of  the 
world.  What  is  known  in  mathematics  as  transforma- 
tion —  that  is,  lawful  transfer  of  attention,  serving  to 
match  in  orderly  fashion  the  things  of  one  system  with 
those  of  another  —  is  conceived  in  life  as  a  process  of 
transmutation  by  which,  in  the  flux  of  the  world,  the 
content  of  the  present  has  come  out  of  the  past  and  in  its 
turn,  in  ceasing  to  be,  gives  birth  to  its  successor,  as 
the  boy  is  father  to  the  man  and  as  things,  in  general, 
become  what  they  are  not.  The  mathematical  concept 
of  invariance  and  that  of  infinitude,  especially  the  im- 
posing doctrines  that  explain  their  meanings  and  bear 
their  names  —  what  are  they  but  mathematicizations 
of  that  which  has  ever  been  the  chief  of  life's  hopes 
and  dreams,  of  that  which  has  ever  been  the  object  of 
its  deepest  passion  and  of  its  dominant  enterprise,  I 
mean  the  finding  of  worth  that  abides,  the  finding  of 
permanence  in  the  midst  of  change,  and  the  discovery 
of  the  presence,  in  what  has  seemed  to  be  a  finite  world, 
of  being  that  is  infinite?  It  is  needless  further  to  mul- 
tiply examples  of  a  correlation  that  is  so  abounding  and 
complete  as  indeed  to  suggest  a  doubt  which  is  the 
juster,  to  view  mathematics  as  the  abstract  idealiza- 
tion of  life,  or  to  regard  life  as  the  concrete  realization 
of  mathematics. 


HUMANIZATION   OP  TEACHING   MATHEMATICS  79 

Finally,  I  wish  to  emphasize  the  fact  that  the  great 
concepts  out  of  which  the  so-called  higher  mathematical 
branches  have  grown  —  the  concepts  of  variable  and 
constant,  of  function,  class  and  relation,  of  transforma- 
tion, invariance,  and  group,  of  finite  and  infinite,  of 
discreteness,  limit,  and  continuity  —  I  wish,  in  closing, 
to  emphasize  the  fact  that  these  great  ideas  of  the 
higher  mathematics,  besides  penetrating  life,  as  we  have 
seen,  in  all  its  complexity  and  all  its  dimensions,  are 
omnipresent,  from  the  very  beginning,  in  the  elements 
of  mathematics  as  well.  The  notion  of  group,  for  ex- 
ample, finds  easy  and  beautiful  illustration,  not  only 
among  the  simpler  geometric  notions  amd  configura- 
tions, but  even  in  the  ensemble  of  the  very  integers 
with  which  we  count.  The  like  is  true  of  the  distinc- 
tion of  finite  and  infinite,  and  of  the  ideas  of  transforma- 
tion, of  invariant,  and  nearly  all  the  rest.  Why  should 
the  presentation  of  them  have  to  await  the  uncertain 
advent  of  graduate  years  of  study?  For  life  already 
abounds,  and  the  great  ideas  that  give  it  its  interests, 
order  and  rationality,  that  is  to  say,  the  focal  concepts 
of  the  higher  mathematics,  are  everywhere  present  in 
the  elements  of  the  science  as  glistening  bassets  of 
gold.  It  is  our  privilege,  in  teaching  the  elements,  to 
avail  ourselves  of  the  higher  conceptions  that  are  present 
in  them;  it  is  our  privilege  to  have  and  to  give  a  lively 
sense  of  their  presence,  their  human  significance,  their 
beauty  and  their  light.  I  do  not  advocate  the  formal 
presentation,  in  secondary  schools,  of  the  higher  con- 
ceptions, in  the  way  of  printed  texts,  for  the  printed 
text  is  apt  to  be  arid  and  the  letter  killeth.  What  I 
wish  to  recommend  is  the  presentation  of  them,  as 
opportunity  may  serve,  in  Greek  fashion,  by  means  of 
dialectic,  face  to  face,  voice  answering  to  voice,  ani- 


80  HUMANIZATION   OF   TEACHING   MATHEMATICS 

mated  with  the  varying  moods  and  motions  and  accents 
of  life  —  laughter,  if  you  will,  and  the  lightning  of  wit 
to  cheer  and  speed  the  slower  currents  of  sober  thought. 
Of  dialectic  excellence,  Plato  at  his  best,  as  in  the 
"Phaedo"  or  the  "Republic,"  gives  us  the  ideal  model 
and  eternal  type.  But  Plato's  ways  are  frequently 
circuitous,  wearisome  and  long.  They  are  ill  suited  to 
the  manners  of  a  direct  and  undeliberate  age;  and  we 
must  find,  each  for  himself,  a  shorter  course.  Somebody 
imbued  with  the  spirit  of  the  matter,  possessed  of  ample 
knowledge  and  having,  besides,  the  requisite  skill  and 
verve  ought  to  write  a  book  showing,  in  so  far  as  the 
printed  page  can  be  made  to  show,  how  naturally  and 
swiftly  and  with  what  a  delightful  sense  of  emancipation 
and  power  thought  may  pass  by  dialectic  paths  from  the 
traditional  elements  of  mathematics  both  to  its  larger 
concepts  and  to  a  vision  of  their  bearings  on  the  higher 
interests  of  life.  I  need  not  say  that  such  a  handling 
of  ideas  implies  much  more  than  a  verbal  knowledge  of 
their  definitions.  It  implies  familiarity  with  the  doc- 
trines that  unfold  the  meanings  of  the  ideas  defined.  It 
is  evident  that,  in  respect  of  this  matter,  the  scripture 
must  read:  Knowing  the  doctrine  is  essential  to  living 
the  life. 


THE  WALLS  OF  THE  WORLD:  OR  CONCERN- 
ING THE  FIGURE  AND  THE  DIMENSIONS 
OF  THE  UNIVERSE  OF  SPACE1 

THERE  is  something  a  little  incongruous  in  attempt- 
ing to  consider  the  subject  of  this  address  in  a  theater 
or  lecture  hall  whose  roof  and  walls  shut  out  from  view 
the  wide  expanses  of  the  world  and  the  azure  deeps. 
For  how  can  we,  amid  the  familiar  finite  scenes  of  a 
closed  and  blinded  room,  command  a  fitting  mood  for 
contemplating  the  infinite  scenes  without  and  beyond? 
A  subject  that  has  sheer  vastness  for  its  central  or 
major  theme  demands  for  its  appropriate  contemplation 
the  still  expanse  of  some  vast  and  open  solitude,  such 
as  the  peak  of  a  lone  and  lofty  mountain  would  afford, 
where  the  gaze  meets  no  wall  save  the  far  horizon  and 
no  roof  but  the  starry  sky.  Perhaps  you  will  be  good 
enough  for  the  time  to  transport  yourselves,  in  imagina- 
tion, into  the  stillness  of  such  a  solitude,  so  that  in  the 
musing  spirit  of  the  place  the  questions  to  be  propounded 
for  consideration  here  may  arise  naturally  and  give  us 
a  due  sense  of  their  significance  and  impressiveness. 
What  are  the  dimensions  and  what  is  the  figure  of  our 
universe  of  space?  How  big  is  it  and  what  is  its  shape? 
What  is  the  figure  of  it  and  what  is  its  size? 

1  An  address  delivered  under  the  auspices  of  the  local  chapters  of  the 
Society  of  Sigma  Xi  at  the  state  universities  of  Minnesota,  Nebraska  and 
Iowa,  April  24,  28  and  30,  1913,  respectively,  and  at  a  joint  meeting  of  the 
chapters  of  Sigma  Xi  and  Phi  Beta  Kappa  of  Columbia  University,  May  8, 
1913.  Printed  in  Science,  June  13,  1913. 


82  THE  WALLS  OF  THE  WORLD 

I  do  not  mind  owning  that  these  questions  have 
haunted  me  a  good  deal  from  the  days  of  my  youth.  It 
happened  in  those  days,  though  I  was  not  aware  of  it 
nor  became  aware  of  it  till  after  many  years,  that  there 
were  then  coming  into  mathematics,  just  entering  the 
fringe,  so  to  speak,  or  the  vestibule  of  the  science,  certain 
striking  ideas  which,  as  I  venture  to  "hope  we  may  see, 
were  destined,  if  not  indeed  to  enable  us  to  answer  the 
questions  with  certainty,  at  all  events  to  clarify  them, 
to  enrich  their  meaning  and  to  make  it  possible  to 
discuss  them  profitably.  It  has  not  been  my  fortune 
to  meet  many  persons  who  had  seriously  propounded 
the  questions  to  themselves  or  who  seemed  to  be  imme- 
diately interested  in  them  when  propounded  by  others 
—  not  many,  even  among  astronomers,  whose  minds, 
it  may  be  assumed,  are  especially  "accustomed  to  con- 
templation of  the  vast."  And  so  I  have  been  forced 
to  the  somewhat  embarrassing  conclusion  that  my  own 
long  interest  in  the  questions  has  been  due  to  the  fact 
of  my  being  of  a  specially  practical  turn  of  mind.  Quite 
seriously  I  venture  to  say  that  we  are  here  engaged  in 
a  practical  enterprise.  .  For  even  if  the  questions  were 
in  the  nature  of  the  case  unanswerable,  which  we  do 
not  admit,  who  does  not  know  how  great  the  boons 
that  have  come  to  men  through  pursuit  of  the  unat- 
tainable? And  who  does  not  know  that,  as  Mr.  Chester- 
ton has  said,  if  you  wish  really  to  know  a  man,  the 
most  practical  question  to  ask  is,  not  about  his  occupa- 
tion or  his  club  membership  or  his  party  or  church 
affiliations,  but  what  are  his  views  of  the  all-embracing 
world?  What  does  he  think  of  the  universe.  Do  but 
fancy  for  a  moment  that  in  somewise  men  should  come 
to  know  the  exact  shape  or  figure  and  especially  the 
exact  size  or  dimensions  of  the  all-immersing  space  of 


THE   WALLS   OF  THE   WORLD  83 

our  universe.  It  requires  but  little  imagination,  not 
much  reflection,  no  extensive  knowledge  of  cosmogonic 
history  and  speculation,  no  very  profound  insight  into 
the  ways  of  truth  to  men,  it  needs,  I  say,  but  little 
philosophic  sense  to  see  that  such  knowledge  would  in 
a  thousand  ways,  direct  and  indirect,  react  powerfully 
upon  our  whole  intelligence,  upon  all  our  attitudes, 
sentiments  and  views,  transforming  our  theology,  our 
ethics,  our  art,  our  religion,  our  philosophy,  our  liter- 
ature, our  science,  and  therewith  affecting  profoundly 
the  whole  sense  and  manner,  the  tone,  color  and  mean- 
ing, of  all  our  institutions  and  the  affairs  of  daily  life. 
Nothing  is  quite  so  practical,  in  the  sense  of  being 
effectual  and  influential,  as  the  views  men  hold,  con- 
sciously or  unconsciously,  regarding  the  great  locus 
of  their  lives  and  their  cosmic  home. 

In  order  to  discuss  the  questions  before  us  intelligibly 
and  profitably  it  is  not  necessary  by  way  of  clearing 
the  ground  to  enter  far  into  metaphysical  speculation 
or  into  psychological  analysis  with  a  view  to  ascer- 
taining what  it  is  that  we  mean  or  ought  to  mean  by 
space.  We  are  not  obliged  to  dispute,  much  less  decide, 
whether  space  is  subjective  or  objective  or  both  or 
indeed  something  that,  as  Plato  in  the  "Timaeus" 
acutely  contends,  is  neither  the  one  nor  the  other.  We 
may  or  may  not  agree  with  the  contention  of  Kant  that 
space  is,  not  an  object,  but  the  form,  of  outer  sense; 
we  may  or  may  not  agree  with  the  radically  different 
contention  of  PoincarS  that  (geometric  as  distinguished 
from  sensible)  space  is  nothing  but  what  is  known  in 
mathematics  as  a  group,  of  which  the  concept  "is  im- 
posed on  us,  not  as  form  of  our  sense,  but  as  form  of 
our  understanding."  It  is,  I  say,  not  necessary  for  us, 
in  tke  interest  of  soundness  and  intelligibility,  to  try 


84  THE  WALLS  OF  THE  WORLD 

to  compose  such  differences  or  to  attempt  a  settlement 
of  these  profound  and  important  questions.  As  to  the 
distinction  between  sensible  space  and  geometric  space, 
it  would  indeed  be  indispensable  to  draw  it  sharply 
and  to  keep  it  always  in  mind,  if  we  were  undertaking 
to  ascertain  what  the  subject  (or  the  object)  of  geom- 
etry is,  or,  what  is  tantamount,  if  we  were  seeking  to 
get  clearly  aware  of  what  it  is  that  geometry  is  about. 
But  in  discussing  the  subject  before  us  it  is  unnecessary 
to  be  always  guarding  that  distinction;  for,  whilst  it 
is  the  space  of  geometry,  and  not  sensible  space,  that 
we  shall  be  talking  about,  yet  it  would  be  a  hindrance 
rather  than  a  help  if  we  did  not  allow,  as  we  habitually 
do  allow,  the  two  varieties  of  space  —  the  imagery  of 
the  one,  the  conceptual  characters  of  the  other  —  to 
mingle  freely  in  our  thinking.  There  will  be  finesse 
enough  for  the  keenest  arrows  of  our  thought  without 
our  going  out  of  the  way  to  find  it.  A  procedure  less 
sophisticated  will  suffice.  It  will  be  sufficient  to  regard 
space  as  being  what,  to  the  layman  and  to  the  student 
of  natural  science,  it  has  always  seemed  to  be:  a  vast 
region  or  room  round  about  us,  an  immense  exteriority, 
locus  of  all  suspended  and  floating  objects  of  outer  sense, 
the  whence,  where  and  whither  of  motion,  theater,  in  a 
word,  of  the  ageless  drama  of  the  physical  universe. 
In  naturally  so  construing  the  term  we  do  not  commit 
ourselves  to  the  philosophy,  so-called,  of  common  sense; 
we  thus  merely  save  our  discourse  from  the  encumbrance 
of  needless  refinements;  for  it  is  obvious  that,  if  space 
be  not  indeed  what  we  have  said  it  seems  to  be,  the 
seeming  is  yet  a  fact,  and  our  questions  would  remain 
without  essential  change:  what,  then,  we  should  ask,  are 
the  dimensions  and  what  is  the  figure  of  that  seeming? 
Though  all  the  things  contained  within  that  triply 


THE  WALLS  OF  THE  WORLD  85 

extended  spread  or  expanse  which  we  call  space  are 
subject  to  the  law  of  ceaseless  change,  the  expanse 
itself,  the  container  of  all,  appears  to  suffer  no  vari- 
ation whatever,  but  to  be,  unlike  time,  a  genuine 
constant,  the  same  yesterday,  today  and  forever,  sole 
absolute  invariant  under  the  infinite  host  of  trans- 
formations that  constitute  the  cosmic  flux.  Whether 
it  be  so  in  fact,  of  course  we  do  not  know.  We  only 
know  that  no  good  reason  has  ever  been  advanced  for 
holding  the  contrary  as  an  hypothesis. 

And  yet  there  is  a  sense,  which  we  ought  I  think  to 
notice,  an  interesting  sense,  in  which  space  seems  to  be, 
not  a  constant,  but,  like  time,  a  variable.  There  is  a 
sense,  deeper  and  juster  perhaps  than  at  first  we  suspect, 
in  which  the  space  of  our  universe  has  in  the  course  of 
time  alternately  shrunken  and  grown.  During  the  last 
century,  for  example,  it  has,  so  it  seems,  greatly  grown, 
in  response,  it  may  be,  to  an  increasing  need  of  the 
human  mind.  By  grown  I  do  not  mean  grown  in  the 
usual  sense,  I  do  not  mean  the  biological  sense,  I  do  not 
mean  the  sense  that  was  present  to  the  mind  of  that 
great  man  Leonardo  da  Vinci,  when  he  wrote  in  effect 
as  follows:  if  you  wish  to  know  that  the  earth  has  been 
growing,  you  have  only  to  observe  "how,  among  the 
high  mountains,  the  walls  of  ancient  and  ruined  cities 
are  being  covered  over  and  concealed  by  the  earth's 
increase";  and,  if  you  would  learn  how  fast  the  earth 
is  growing,  you  have  only  to  set  a  vase,  filled  with  pure 
earth,  upon  a  roof;  to  note  how  green  herbs  will  imme- 
diately begin  to  shoot  up;  to  note  that  these,  when 
mature,  will  cast  their  seeds;  to  allow  the  process  to 
continue  through  repetition;  then,  after  the  lapse  of  a 
decade,  to  measure  the  soil's  increase;  and,  finally, 
to  multiply,  in  order  to  have  thus  determined  "how 


86  THE  WALLS  OF  THE  WORLD 

much  the  earth  has  grown  in  the  course  of  a  thousand 
years."  In  this  matter,  Leonardo  was  doubtless  wrong. 
At  all  events  current  scientific  views  are  against  him. 
The  earth,  we  know,  has  grown,  but  the  growth  has 
been  by  accretion,  by  addition  from  without,  and  not, 
in  biologic  sense,  by  expansion  from  within  (unless, 
indeed,  we  adopt  the  beautiful  hypothesis  of  the  poet 
and  physicist,  Theodor  Fechner,  for  which  so  hard- 
headed  a  scientific  man  as  Bernhardt  Riemann  had  so 
much  respect,  the  hypothesis,  namely,  that  the  plants, 
the  earth  and  the  stars  have  souls).  The  myriad- 
minded  Florentine  was,  we  of  today  think,  in  error, 
his  error  being  one  of  those  brilliant  mistakes  that  but 
few  men  have  been  qualified  to  make.  But  in  saying 
that  space  has  grown  we  do  not  mean  that  it  has  grown 
in  the  biologic  sense  of  Leonardo  nor  yet  in  the  sense 
of  addition  from  without.  We  mean  that  it  has  grown 
as  a  thing  in  mind  may  grow,  as  a  thing  in  thought 
may  grow;  we  mean  that  it  has  grown  in  men's  con- 
ception of  it.  That  space  has,  in  this  sense,  been  en- 
larged prodigiously  in  the  course  of  recent  time  is  evident 
to  all.  It  has  been  often  said  that  "  the  first  grand  dis- 
covery of  modern  times  is  the  immense  extension  of  the 
universe  in  space."  It  would  be  juster  to  say  that  the 
first  grand  achievement  of  modern  science  has  been 
the  immense  extension  of  space  itself,  the  prodigious 
enlargement  of  it,  in  the  imagination  and  especially  in 
the  thought  of  men.  If  we  will  but  take  the  trouble 
to  recall  vividly  the  Mosaic  cosmogony,  in  terms  of 
which  most  of  us  have  but  recently  ceased  to  frame  our 
sublimest  conceptions  of  the  vast:  if  we  remind  our- 
selves of  Plato's  "concentric  crystal  spheres,  the  ada- 
mantine axis  turning  in  the  lap  of  necessity,  the  bands 
that  held  the  heaven  together  like  a  girth  that  clasps 


THE  WALLS  OF  THE  WORLD  87 

a  ship,  the  shaft  which  led  from  earth  to  sky,  and  which 
was  paced  by  the  soul  in  a  thousand  years";  if  we 
compare  these  conceptions  with  our  own;  if  we  think 
of  "the  fields  from  which  our  stars  fling  us  their  light," 
fields  that  are  really  near  and  yet  are  so  far  that  the 
swiftest  of  messengers,  capable  of  circling  the  earth 
eight  times  in  a  second,  requires  for  its  journey  hither 
thousands  of  years;  if  we  do  but  make  some  such  com- 
parisons, we  shall  begin  to  realize  dimly  that,  compared 
with  modern  space  —  the  space  of  modern  thought  — 
elder  space  —  the  space  of  elder  thought  —  is  indeed 
"but  as  a  cabinet  of  brilliants,  or  rather  a  little  jewelled 
cup  found  in  the  ocean  or  the  wilderness." 

Suppose  that  in  fact  space  were  thus,  like  time,  not  a 
constant,  but  a  variable;  suppose  it  were  a  mental  thing 
growing  with  the  growth  of  mind;  an  increasing  function 
of  increasing  thought;  suppose  it  were  a  thing  whose 
enlargement  is  essential  as  a  psychic  condition  or  con- 
comitant or  effect  of  the  progress  of  science;  would  not 
our  questions  regarding  its  figure  and  its  dimensions 
then  lose  their  meaning?  The  answer  is,  no;  as  rational 
beings  we  should  still  be  bound  to  ask:  what  are  the 
dimensions  and  what  is  the  figure  of  space  to  date? 
That  is  not  all.  If  these  questions  were  answered,  we 
could  propound  the  further  questions:  whether  the 
space  so  characterized  —  the  space  of  the  present  —  is 
adequate  to  the  present  needs  of  science,  and  whether 
it  is  not  destined  to  yet  further  expansion  in  response 
to  the  future  needs  of  thought. 

Men  do  not  feel,  however,  that  such  spatial  enlarge- 
ments as  I  have  indicated  are  genuine  enlargements  of 
space.  In  spite  of  whatever  metaphysics  or  psychology 
may  seem  obliged  to  say  to  the  contrary,  men  feel  that 
what  is  nrw  in  such  an  enlargement  is  merely  an  in- 


88  THE  WALLS  OF  THE  WORLD 

crease  of  enlightenment  regarding  something  old;  they 
feel  that  what  is  new  is,  not  an  added  vastness,  but  a 
discovery  of  a  vastness  that  always  was  and  always  will 
be.  Let  us  trust  this  feeling  and,  regarding  space  as 
constant  from  everlasting  to  everlasting,  let  us  take  the 
questions  in  their  natural  intent  and  form:  what  are 
the  dimensions  and  what  is  the  figure  of  our  universe 
of  space? 

If  you  propound  these  questions  to  a  normal  student 
of  natural  science,  say  to  a  normal  astronomer,  his  re- 
sponse will  be  —  what?  If  you  appear  to  him  to  be 
quite  sincere  and  if,  besides,  he  be  in  an  amiable  mood, 
his  response  will,  not  improbably,  be  a  significant  shrug 
of  the  shoulders,  designed  to  intimate  that  his  time  is 
too  precious  to  be  squandered  in  considering  questions 
that,  if  not  meaningless,  are  at  all  events  unanswerable. 
I  maintain,  on  the  contrary,  that  this  same  student  of 
natural  science,  and  indeed,  all  other  normally  educated 
men  and  women,  have,  as  a  part  of  their  intellectual 
stock  in  trade,  perfectly  definite  answers  to  both  of  the 
questions.  I  do  not  mean  that  they  are  aware  of  pos- 
sessing such  wealth  nor  shall  I  undertake  to  say  in 
advance  whether  their  answers  be  correct.  What  I 
am  asserting  and  what,  with  your  assistance,  I  shall 
endeavor  to  demonstrate,  is  that  perfectly  precise, 
very  intelligent  and  perfectly  intelligible  answers  to 
both  of  the  questions  are  logically  involved  in  what 
every  normally  educated  mind  regards  as  the  securest 
of  its  intellectual  possessions.  In  order  to  show  that 
such  answers  are  to  be  found  embedded  in  the  content 
of  the  normally  educated  mind  and  in  order  to  lay  them 
bare,  it  will  be  necessary  to  have  recourse  to  the  process 
of  explication.  Explication,  however,  is  nothing  strange 
to  an  academic  audience.  It  is  true,  indeed,  that  we 


THE    WALLS   OF   THE   WORLD  89 

no  longer  derive  the  verb,  to  educate,  from  educere,  but 
it  is  yet  a  fact,  as  every  one  knows,  that  a  large  part 
of  education  is  eduction  —  the  leading  forth  into  light 
what  is  hidden  in  the  familiar  content  of  our  minds. 

What  are  those  answers?  I  shall  present  them  in  the 
familiar  and  brilliant  words  of  one  who  in  the  span  of  a 
short  life  achieved  a  seven-fold  immortality:  immortality 
as  a  physicist,  as  a  philosopher,  as  a  mathematician,  as 
a  theologian,  as  a  writer  of  prose,  as  an  inventor  and  as 
a  fanatic.  From  this  brief  but  "immortal"  characteriza- 
tion I  have  no  doubt  that  you  detect  the  author  at 
once  and  at  once  recall  the  words:  Space  is  an  infinite 
sphere  whose  center  is  everywhere  and  whose  surface  is 
nowhere. 

You  will  observe  that,  without  change  of  meaning, 
I  have  substituted  "space"  for  "universe"  and  "sur- 
face" for  "circumference."  This  brilliant  mot  of  Blaise 
Pascal,  as  every  one  knows,  has  long  been  valued 
throughout  the  world  as  a  splendid  literary  gem.  I 
am  not  aware  that  it  has  been  at  any  time  regarded 
seriously  as  a  scientific  thesis.  It  may,  however,  be 
so  regarded.  I  propose  to  show,  with  your  co-opera- 
tion, that  this  exquisite  saying  of  Pascal  expresses  with 
mathematical  precision  the  firm,  albeit  unconscious,  con- 
viction of  the  normally  educated  mind  respecting  the 
size  and  the  shape  of  the  space  of  our  universe.  Be 
good  enough  to  note  carefully  at  the  outset  the  car- 
dinal phrases:  infinite  sphere,  center  everywhere,  surface 
nowhere. 

If  you  are  told  that  there  is  an  object  completely 
enclosed  and  that  the  object  is  equally  distant  from  all 
parts  of  the  enclosing  boundary  or  wall,  you  instantly 
and  rightly  think  of  a  sphere  having  that  object  as 
center.  Let  me  ask  you  to  think  of  some  point,  any 


90  THE  WALLS  OF  THE  WORLD 

convenient  point,  P,  together  with  all  the  straight  lines 
or  rays  —  called  a  sheaf  of  lines  or  rays  —  that,  begin- 
ning at  P,  run  out  from  it  as  far  as  ever  the  nature  of 
space  allows.  We  ask:  do  all  the  rays  of  the  sheaf 
run  out  equally  far?  It  seems  perfectly  evident  that  they 
do,  and  with  this  we  might  be  content.  It  will  be  worth 
while,  however,  to  examine  the  matter  a  little  more 
attentively.  Denote  by  L  any  chosen  line  or  ray  of  the 
sheaf.  Choose  any  convenient  unit  of  length,  say  a 
mile.  We  now  ask:  how  many  of  our  units,  how  many 
miles  can  we,  starting  from  P,  lay  off  along  L?  Lay 
off,  I  mean,  not  in  fact,  but  in  thought.  In  other 
words :  how  many  steps,  each  a  mile  long,  can  we, 
in  traversing  L,  take  in  thought?  Hereafter  let  the 
phrase  "in  thought"  be  understood.  Can  the  question 
be  answered?  It  can.  .  Can  it  be  answered  definitely? 
Absolutely  so.  How?  As  follows.  Before  proceeding, 
however,  let  me  beg  of  you  not  to  hesitate  or  shy  if  cer- 
tain familiar  ideas  seem  to  get  submitted  to  the  logical 
process — the  mind-expanding  process  —  of  generalization. 
There  is  to  be  no  resort  to  any  kind  of  legerdemain. 
Let  us  be  willing  to  transcend  imagination,  and,  with- 
out faltering,  to  follow  thought,  for  thought,  free  as  the 
spirit  of  creation,  owns  no  bar  save  that  of  inconsistence 
or  self-contradiction.  Consider  the  sequence  of  cardinal 
numbers, 

(S)     i,  2,  3,  4,  5,  6,  7,  .... 

The  sequence  is  neither  so  dry  nor  so  harmless  as  it 
seems.  It  has  a  beginning;  but  it  has  no  end,  for,  by 
the  law  of  its  formation,  after  each  term  there  is  a  next. 
The  difference  between  a  sequence  that  stops  somewhere 
and  one  that  has  no  end  is  awful.  No  one,  unless  spir- 
itually unborn  or  dead,  can  contemplate  that  gulf 
without  emotions  that  take  hold  of  the  infinite  and 


THE  WALLS  OF  THE  WORLD  91 

everlasting.  Let  us  compare  the  sequence  with  the  ray 
L  of  our  sheaf.  Choose  in  (5)  any  number  n,  however 
large.  Can  we  go  from  P  along  L  that  number  n 
of  miles?  We  are  certain  that  we  can.  Suppose  the 
trip  made,  a  mile  post  set  up  and  on  it  painted  the 
number  n  to  tell  how  far  the  post  is  from  P.  As  n  is 
any  number  in  (5),  we  may  as  well  suppose,  indeed  we 
have  already  implicitly  supposed,  mile  posts,  duly  dis- 
tributed and  marked,  to  have  been  set  up  along  L  to 
match  each  and  every  number  in  the  sequence.  Have 
we  thus  set  up  all  the  mile  posts  that  L  allows?  We 
are  certain  that  we  have,  for,  if  we  go  out  from  P  along 
L  any  possible  but  definite  number  of  miles,  we  are 
perfectly  certain  that  that  number  is  a  number  in  the 
sequence,  and  that  accordingly  the  journey  did  but  take 
us  to  a  post  set  up  before.  What  is  the  upshot?  It 
is  that  L  admits  of  precisely  as  many  mile  posts  as 
there  are  cardinal  numbers,  neither  more  nor  less.  How 
long  is  L?  The  answer  is:  L  is  exactly  as  many  miles 
long  as  there  are  integers  or  terms  in  the  sequence  (5). 
Can  we  say  of  any  other  line  or  ray  L'  of  the  sheaf 
what  we  have  said  of  L?  We  are  certain  that  we  can. 
Indeed  we  have  said  it,  for  L  was  any  line  of  the  sheaf. 
May  we,  then,  say  that  any  two  lines,  L  and  L',  of  the 
sheaf  are  equal?  We  may  and  we  must.  For,  just  as 
we  have  established  a  one-to-one  correspondence  between 
the  mile  posts  of  L  and  the  terms  of  (5),  so  we  may 
establish  a  one-to-one  correspondence  between  the  mile 
posts  of  L  and  those  of  L't  and  what  we  mean  by  the 
equality  of  two  classes  of  things  is  precisely  the  possi- 
bility of  thus  setting  up  a  one-to-one  correlation  be- 
tween them.  Accordingly,  all  the  lines  or  rays  of  our 
sheaf  are  equal.  We  can  not  fail  to  note  that  thus 
there  is  forming  in  our  minds  the  conception  of  a  sphere, 


Q2  THE  WALLS  OF  THE  WORLD 

centered  at  P,  larger,  however,  than  any  sphere  of  slate 
or  wood  or  marble  —  a  sphere,  if  it  be  a  sphere,  whose 
radii  are  the  rays  of  our  sheaf.  Is  not  the  thing,  how- 
ever, too  vast  to  be  a  sphere?  Obviously  yes,  if  the 
lines  or  rays  of  the  sheaf  have  a  length  that  is  indefinite, 
unassignable;  obviously  no,  if  their  length  be  assignable 
and  definite.  We  have  seen  the  length  of  a  ray  contains 
exactly  as  many  miles  as  there  are  integers  or  terms 
in  (S).  The  question,  then,  is:  has  the  totality  of  these 
terms  a  definite  assignable  number?  The  answer  is, 
yes.  To  show  it,  look  sharply  at  the  following  fact, 
a  bit  difficult  to  see  only  because  it  is  so  obvious,  being 
writ,  so  to  speak,  on  the  very  surface  of  the  eye.  I 
wish,  in  a  word,  to  make  clear  what  is  meant  by  the 
cardinal  number  of  any  given  class  of  things.  The 
fingers  of  my  right  hand  constitute  a  class  of  objects; 
the  fingers  of  my  left  hand,  another  class.  We  can  set 
up  a  one-to-one  correspondence  between  the  classes, 
pairing  the  objects  in  the  one  with  those  in  the  other. 
Any  two  classes  admitting  of  such  a  correlation  are 
said  to  be  equivalent.  Now  given  any  class  K,  there  is 
another  class  C  composed  of  all  the  classes  each  of 
which  is  equivalent  to  K.  C  is  called  the  cardinal  num- 
ber of  K,  and  the  name  of  C,  if  it  has  received  a  name, 
tells  how  many  objects  are  in  K.  Thus,  if  K  is  the  class 
of  the  fingers  of  my  right  hand,  the  word  five  is  the  name 
of  the  class  of  classes  each  equivalent  to  K.  Now  to 
the  application.  The  terms  of  (S)  constitute  a  class  K 
(of  terms).  Has  it  a  definite  number?  Yes.  What  is 
it?  It  is  the  class  of  all  classes  each  equivalent  to  K. 
Has  this  number  class  received  a  name  of  its  own?  Yes, 
and  it  has,  like  many  other  numbers,  received  a  symbol, 
namely,  X,  read  Aleph  null.  It  is,  then,  this  cardinal 
number  Aleph,  not  familiar,  indeed,  but  perfectly  definite 


THE   WALLS   OF  THE   WORLD  93 

as  denoting  a  definite  class,  it  is  this  that  tells  us  how 
many  terms  are  in  (5)  and  therewith  tells  us  the  length 
of  the  rays  of  our  sheaf.  Herewith  the  concept  that 
was  forming  is  now  completely  formed:  space  is  a  sphere 
centered  at  P. 

But  is  the  sphere,  as  Pascal  asserts,  an  infinite  sphere? 
We  may  easily  see  that  it  is.  Again  consider  the  se- 
quence (S)  and  with  it  the  similar  sequence  (50, 


(S)     i,  2,  3,  4,  s,  6,  7,  •  .  •, 
(50    a,  4,  6,  8,  10,  12,  14,  .. 


Observe  that  all  the  terms  in  (S')  are  in  (5)  and  that 
(S)  contains  terms  that  are  not  in  (Sf).  (S')  is,  then, 
a  proper  part  of  (5).  Next  observe  that  we  can  pair 
each  term  in  (5)  with  the  term  below  it  in  (S7).  That 
is  to  say:  the  whole,  (5),  is  equivalent  to  one  of  its 
parts,  (5').  A  class  that  thus  has  a  part  to  which  it 
is  equivalent  is  said  to  be  infinite,  and  the  number  of 
things  in  such  a  class  is  called  an  infinite  number.  Aleph 
is,  then,  an  infinite  number,  and  so  we  see  that  the  rays 
of  our  sheaf,  the  radii  of  our  sphere,  are  infinite  in 
length:  space  is  an  infinite  sphere  centered  at  P. 

Finally,  what  of  the  phrases,  center  everywhere,  surface 
nowhere?  Can  we  give  them  a  meaning  consistent  with 
common  usage  and  common  sense?  We  can,  as  follows. 
Let  O  be  any  chosen  point  somewhere  in  your  neigh- 
borhood. By  saying  that  the  center  P  is  everywhere 
we  mean  that  P  may  be  taken  to  be  any  point  within 
a  sphere  centered  at  O  and  having  a  finite  radius,  a 
radius,  that  is,  whose  length  in  miles  is  expressed  by  any 
integer  in  (S).  And  by  saying  that  the  surface  of  our 
infinite  sphere  is  nowhere  we  mean  that  no  point  of  the 
surface  can  be  reached  by  traveling  out  from  P  any 
finite  number,  however  large,  of  miles,  by  traveling,  that 


94  THE  WALLS  OF  THE  WORLD 

is,  a  number  of  miles  expressed  by  any  number,  however 
large,  in  (5). 

Here  we  have  touched  our  primary  goal:  we  have 
demonstrated  that  men  and  women  whose  education, 
in  respect  of  space,  has  been  of  normal  type,  believe 
profoundly,  albeit  unawares,  that  the  space  of  our  uni- 
verse is  an  infinite  sphere  of  which  the  center  is  every- 
where and  the  surface  nowhere.  Such  is  the  beautiful 
conception,  the  great  conception  — •  mathematically  pre- 
cise yet  mystical  withal  and  awful  in  its  limitless  reaches 
—  which  is  ever  ready  to  form  itself,  in  the  normally 
educated  mind  and  there  to  stand  a  deep-rooted  con- 
scious conviction  regarding  the  shape  and  the  size 
of  the  all-embracing  world. 

Is  the  conception  valid?  Does  the  conviction  corre- 
spond to  fact?  Is  it  true?  It  is  not  enough  that  it  be 
intelligible,  which  it  is;  it  is  not  enough  that  it  be  noble 
and  sublime,  which  also  it  is.  No  doubt  whatever  is 
noble  and  sublime  is,  in  some  sense,  true.  For  we 
mortals  have  to  do  with  more  than  reason.  Yet  science, 
science  in  the  modern  technical  sense  of  the  term,  having 
elected  for  its  field  the  domain  of  the  rational,  allows 
no  superrational  tests  of  truth  to  be  sufficient  or  final. 
We  must,  therefore,  ask:  are  the  dimensions  and  the 
figure  of  our  space,  in  fact,  what,  as  we  have  seen, 
Pascal  asserts  and  the  normally  educated  mind  believes 
them  to  be?  Long  before  the  days  of  Pascal,  back 
yonder  in  the  last  century  before  the  beginning  of  the 
Christian  era,  one  of  the  acutest  and  boldest  thinkers 
of  all  time,  immortal  expounder  of  Epicurean  thought, 
answered  the  question,  with  the  utmost  confidence,  in 
the  affirmative.  I  refer  to  Lucretius  and  his  "De  Rerum 
Natura."  In  my  view  that  poem  is  the  greatest  and 
finest  union  of  literary  excellence  and  scientific  spirit  to 


THE   WALLS  OF  THE   WORLD  95 

be  found  in  the  annals  of  human  thinking.  I  main- 
tain that  opinion  of  the  work  despite  the  fact  that  the 
majority  of  its  conclusions  have  been  invalidated  by 
time,  have  perished  by  supersession;  for  we  must  not 
forget  that,  in  respect  of  knowledge,  "the  present  is 
no  more  exempt  from  the  sneer  of  the  future  than  the 
past  has  been."  I  maintain  that  opinion  of  the  work 
despite  the  fact  that  the  enterprise  of  Lucretius  was 
marvelously  extravagant;  for  we  must  not  forget  that 
the  relative  modesty  of  modern  men  of  science  is  not 
inborn,  but  is  only  an  imperfectly  acquired  lesson. 
Well,  it  is  in  that  great  work  that  Lucretius  endeavors 
to  prove  that  our  universe  of  space  is  infinite  in  the 
sense  that  we  have  explained.  His  argument,  which 
runs  to  many  words,  may  be  briefly  paraphrased  as 
follows.  Conceive  that,  starting  from  any  point  of 
space,  you  go  out  in  any  direction  as  far  as  you  please, 
and  that  then  you  hurl  your  javelin.  Either  it  will 
go  on,  in  which  case  there  is  space  ahead  for  it  to  move 
in,  or  it  will  not  go  on,  in  which  case  there  must  be 
space  ahead  to  contain  whatever  prevents  its  going. 
In  either  case,  then,  however  far  you  may  have  gone, 
there  is  yet  space  beyond.  And  so,  he  concludes,  space 
is  infinite,  and  he  triumphantly  adds: 

Therefore  the  nature  of  room  and  the  space  of  the  unfathomable  void 
are  such  as  bright  thunderbolts  can  not  race  through  in  their  course  though 
gliding  on  through  endless  tract  of  time,  no  nor  lessen  one  jot  the  journey 
that  remains  to  go  by  all  their  travel  —  so  huge  a  room  is  spread  out  on 
all  sides  for  things  without  any  bounds  in  all  directions  round. 

Such  is  the  argument,  the  great  argument,  of  the 
Roman  poet.  Great  I  call  it,  for  it  is  great  enough  to 
have  fooled  all  philosophers  and  men  of  science  for  two 
thousand  years'.  Indeed  only  a  decade  ago  I  heard 
the  argument  confidently  employed  by  an  American 


96  THE  WALLS  OF  THE  WORLD 

thinker  of  more  than  national  reputation.  But  is  the 
argument  really  fallacious?  It  is.  The  conclusion  may 
indeed  be  quite  correct  —  space  may  indeed  be  infinite, 
as  Lucretius  asserts  —  but  it  does  not  follow  from  his 
argument.  To  show  the  fallacy  is  no  difficult  feat. 
Consider  a  sphere  of  finite  radius.  We  may  suppose 
it  to  be  very  small  or  intermediate  or  very  large  —  no 
matter  what  its  size  so  long  as  its  radius  is  finite.  By 
sphere,  in  this  part  of  the  discussion,  I  shall  mean  sphere- 
surface.  Be  good  enough  to  note  and  bear  that  in  mind. 
Observe  that  this  sphere  —  this  surface  —  is  a  kind  of 
room.  It  is  a  kind  of  space,  region  or  room  where 
certain  things,  as  points,  circle  arcs  and  countless  other 
configurations  can  be  and  move.  These  things,  con- 
fined to  this  surface,  which  is  their  world,  their  universe 
of  space,  if  you  please,  enjoy  a  certain  amount,  an 
immense  amount,  of  freedom:  the  points  of  this  world 
can  move  in  it  hither,  thither  and  yonder;  they  can 
move  very  far,  millions  and  millions  of  miles,  even  in 
the  same  direction,  if  only  the  sphere  be  taken  large 
enough.  I  see  no  reason  why  we  should  not,  for  the 
sake  of  vividness,  fancy  that  spherical  world  inhabited 
by  two-dimensional  intelligences  conformed  to  their 
locus  and  home  just  as  we  are  conformed  to  our  own 
space  of  three  dimensions.  I  see  no  reason  why  we 
should  not  fancy  those  creatures,  in  the  course  of  their 
history,  to  have  had  their  own  Democritus  and  Epicurus, 
to  have  had  their  own  Roman  republic  or  empire  and 
in  it  to  have  produced  the  brilliant  analogues  of  our  own 
Virgil,  Cicero  and  Lucretius.  Do  but  note  attentively 
—  for  this  is  the  point  —  that  their  Lucretius  could 
have  said  about  their  space  precisely  what  our  own 
said  about  ours.  Their  Lucretius  could  have  said  to 
his  fellow-inhabitants  of  the  sphere:  "starting  at  any 


THE   WALLS   OF   THE   WORLD  97 

point,  go  as  far  as  ever  you  please  in  any  straight  line" 
—  such  line  would  of  course  (as  we  know)  be  a  great 
circle  of  the  sphere  —  "and  then  hurl  your  javelin" 
the  javelin  would,  as  we  know,  be  only  an  arc  of  a  great 
circle  —  "either  it  will  go  on,  in  which  case,  etc.;  or 
it  will  not,  etc.";  thus  giving  an  argument  identical 
with  that  of  our  own  Lucretius.  But  what  could  it 
avail?  We  know  what  would  happen  to  the  javelin 
when  hurled  as  supposed  in  the  surface:  it  would  go 
on  for  a  while,  there  being  nothing  to  prevent  it.  But 
whether  it  went  on  or  not,  it  could  not  be  logically 
inferred  that  the  surface,  the  space  in  question,  is  infinite, 
for  we  know  that  the  surface  is  finite,  just  so  many,  a 
finite  number  of,  square  miles.  The  fallacy,  at  length, 
is  bare.  It  consists  —  the  fact  has  been  recently  often 
pointed  out  —  in  the  age-long  failure  to  distinguish 
adequately  between  unbegrenzt  and  unendlich  —  between 
boundless  and  infinite  as  applied  to  space.  What  our 
fancied  Lucretius  proved  is,  if  anything,  that  the  sphere 
is  boundless,  but  not  that  it  is  infinite.  What  our  real 
Lucretius  proved  is,  if  anything,  that  the  space  of  our 
universe  is  boundless,  but  not  that  it  is  infinite.  That 
a  region  or  room  may  be  boundless  without  being  in- 
finite is  clearly  shown  by  the  sphere  (surface).  How 
evident,  once  it  is  drawn,  the  distinction  is.  And  yet 
it  was  never  drawn,  in  thinking  about  the  dimensions 
of  space,  until  in  1854  it  was  drawn  by  Riemann  in 
his  epoch-marking  and  epoch-making  Habilitalionschrift 
on  the  foundations  of  geometry. 

What,  then,  is  the  fact?  Is  space  finite,  as  Riemann 
held  it  may  be?  Or  is  it  infinite,  as  Lucretius  and  Pascal 
deliberately  asserted,  and  as  the  normally  educated 
mind,  ho  we  vet  unconsciously,  yet  firmly  believes?  No 
one  "knows.  The  question  is  one  of  the  few  great  out- 


98  THE  WALLS  OF  THE  WORLD 

standing  scientific  questions  that  intelligent  laymen  may, 
with  a  little  expert  assistance,  contrive  to  grasp.  Shall 
we  ever  find  the  answer?  Time  is  long,  and  neither 
science  nor  philosophy  feels  constrained  to  haul  down 
the  flag  and  confess  an  ignorabimus.  Neither  is  it 
necessary  or  wise  for  science  and  philosophy  to  camp 
indefinitely  before  a  problem  that  they  are  evidently 
not  yet  equipped  to  solve.  They  may  proceed  to  related 
problems,  always  reserving  the  right  to  return  with 
better  instruments  and  added  light. 

In  the  present  instance,  let  us  suppose,  for  the  mo- 
ment, that  Lucretius,  Pascal  and  the  normally  educated 
mind  are  right:  let  us  suppose  that  space  is  infinite,  as 
they  assert  and  believe.  In  that  case  the  bounds  of  the 
universe  are  indeed  remote,  and  yet  we  may  ask:  are 
there  not  ways  to  pass  in  thought  the  walls  of  even  so 
vast  a  world?  There  are  such  ways.  But  where  and 
how?  For  are  we  not  supposing  that  the  walls  to  be 
passed  are  distant  by  an  amount  that  is  infinite?  And 
how  may  a  boundary  that  is  infinitely  removed  be 
reached  and  overpassed?  The  answer  is  that  there  are 
many  infinites  of  many  orders;  that  infinites  are  sur- 
passed by  other  infinites;  that  infinites,  like  the  stars, 
differ  in  glory.  This  is  not  rhetoric,  it  is  naked  fact. 
One  of  the  grand  achievements  of  mathematics  in  the 
nineteenth  century  is  to  have  defined  infinitude  (as 
above  defined)  and  to  have  discovered  that  infinites 
rise  above  infinites,  in  a  genuine  hierarchy  without  a 
summit.  In  order  to  show  how  we  can  in  thought  pass 
the  Lucretian  and  Pascal  walls  of  our  universe,  I  must 
ask  you  to  assume  as  a  lemma  a  mathematical  proposi- 
tion which  has  indeed  been  rigorously  established  and  is 
familiar,  but  the  proof  of  which  we  can  not  tarry  here 
to  reproduce.  Consider  all  the  real  numbers  from  zero 


THE  WALLS  OF  THE  WORLD  99 

to  one  inclusive,  or,  what  is  tantamount,  consider  all 
the  points  in  a  unit  segment  of  a  continuous  straight 
line.  The  familiar  proposition  that  I  am  asking  you 
to  assume  is  that  it  is  not  possible  to  set  up  a  one-to- 
one  correspondence  between  the  points  of  that  segment 
and  the  positive  integers  (in  the  sequence  above  given), 
but  that,  if  you  take  away  from  the  segment  an  infini- 
tude (Aleph)  of  points  matching  all  the  positive  integers, 
there  will  remain  in  the  segment  more  points,  infinitely 
more,  than  you  have  taken  away.  That  means  that 
the  infinitude  of  points  in  the  segment  infinitely  surpasses 
the  infinitude  of  positive  integers;  surpasses,  that  is,  the 
infinitude  of  mile  posts  on  the  radius  of  our  infinite 
(Pascal)  sphere.  Now  conceive  a  straight  line  containing 
as  many  miles  as  there  are  points  in  the  segment.  You 
see  at  once  that  in  that  conception  you  have  overleaped 
the  infinitely  distant  walls  of  the  Lucretian  universe. 
Overleaped,  did  I  say?  Nay,  you  have  passed  beyond 
those  borders  by  a  distance  infinitely  greater  than  the 
length  of  any  line  contained  within  them.  And  thus 
it  appears  that,  not  our  imagination,  indeed,  but  our 
reason  may  gaze  into  spatial  abysses  beside  which  the 
infinite  space  of  Lucretius  and  Pascal  is  but  a  meager 
thing,  infinitesimally  small.  There  remain  yet  other 
ways  by  which  we  are  able  to  escape  the  infinite  con- 
fines of  this  latter  space.  One  of  these  ways  is  pro- 
vided in  the  conception  of  hyperspaces  enclosing  our 
own  as  this  encloses  a  plane.  But  that  is  another  story, 
and  the  hour  is  spent. 

The  course  we  have  here  pursued  has  not,  indeed, 
enabled  us  to  answer  with  final  assurance  the  two  ques- 
tions with  which  we  set  out.  I  hope  we  have  seen  along 
the  way  something  of  the  possibilities  involved.  I  hope 
we  have  gained  some  insight  into  the  meaning  of  the 


100  THE  WALLS  OF  THE  WORLD 

questions  and  have  seen  that  it  is  possible  to  discuss 
them  profitably.  And  especially  I  hope  that  we  have 
seen  afresh,  what  we  have  always  to  be  learning  again, 
that  it  is  not  in  the  world  of  sense,  however  precious 
it  is  and  ineffably  wonderful  and  beautiful,  nor  yet  in 
the  still  finer  and  ampler  world  of  imagination,  but 
it  is  in  the  world  of  conception  and  thought  that  the 
human  intellect  attains  its  appropriate  freedom  —  a  free- 
dom without  any  limitation  save  the  necessity  of  being 
consistent.  Consistency,  however,  is  only  a  prosaic  name 
for  a  limitation  which,  in  another  and  higher  realm, 
harmony  imposes  even  upon  the  muses. 


MATHEMATICAL  EMANCIPATIONS: 
DIMENSIONALITY  AND  HYPERSPACE1 

AMONG  the  splendid  generalizations  effected  by  modern 
mathematics,  there  is  none  more  brilliant  or  more  in- 
spiring or  more  fruitful,  and  none  more  nearly  commen- 
surate with  the  limitless  immensity  of  being  itself,  than 
that  which  produced  the  great  concept  variously  desig- 
nated by  such  equivalent  terms  as  hyperspace,  multidi- 
mensional space,  n- space,  n-fold  or  w -dimensional  space, 
and  space  of  n  dimensions. 

In  science  as  in  life  the  greatest  truths  are  the  sim- 
plest. Intelligibility  is  alike  the  first  and  the  last  de- 
mand of  the  understanding.  Naturally,  therefore,  those 
scientific  generalizations  that  have  been  accounted  really 
great,  such  as  the  Newtonian  law  of  gravitation,  or  the 
principle  of  the  conservation  of  energy,  or  the  all- 
conquering  concept  of  cosmic  evolution,  are,  all  of  them, 
distinguished  by  their  simplicity  and  apprehensibility. 
To  that  rule  the  notion  of  hyperspace  presents  no  excep- 
tion. For  its  fair  understanding,  for  a  live  sensibility 
to  its  manifold  significance  and  quickening  power,  a 
long  and  severe  mathematical  apprenticeship,  however 
helpful  it  would  be,  is  not  demanded  in  preparation, 
but  only  the  serious  attention  of  a  mature  intelligence 
reasonably  inured  by  discipline  to  the  exactions  of  ab- 
stract thought  and  the  austerities  of  the  higher  imag- 

1  Printed  in  Tke  Monist,  January,  1906.  For  a  deeper  view  of  this  subject 
the  reader  may  be  referred  to  the  ijlh  essay  of  this  volume. 


102  MATHEMATICAL  EMANCIPATIONS 

ination.  And  it  is  to  the  reader  having  this  general 
equipment,  rather  than  to  the  professional  mathe- 
matician as  such,  that  the  present  communication  is 
addressed. 

To  a  clear  understanding  of  what  the  mathematician 
means  by  hyperspace,  it  is  in  the  first  place  necessary 
to  conceive  in  its  full  generality  the  closely  related  notion 
of  dimensionality  and  to  be  able  to  state  precisely  what 
is  meant  by  saying  that  a  given  manifold  has  such  and 
such  a  dimensionality,  or  such  and  such  a  number  of 
dimensions,  in  a  specified  entity  or  element. 

Discrimination,  as  the  proverb  rightly  teaches,  is  the 
beginning  of  mind.  The  first  psychic  product  of  that 
initial  psychic  act  is  numerical:  to  discriminate  is  to 
produce  two,  the  simplest  possible  example  of  multi- 
plicity. The  discovery,  or  better  the  invention,  better 
still  the  production,  best  of  all  the  creation,  of  multi- 
plicity with  its  correlate  of  number,  is,  therefore,  the 
most  primitive  achievement  or  manifestation  of  mind. 
Such  creation  is  the  immediate  issue  of  intellection, 
nay,  it  is  intellection,  identical  with  its  deed,  and,  with- 
out the  possibility  of  the  latter,  the  former  itself  were 
quite  impossible.  Accordingly  it  is  not  matter  for  sur- 
prise but  is  on  the  contrary  a  perfectly  natural  or  even 
inevitable  phenomenon  that  explanations  of  ultimate 
ideas  and  ultimate  explanations  in  general  should  more 
and  more  avail  themselves  of  analytic  as  distinguished 
from  intuitional  means  and  should  tend  more  and  more 
to  assume  arithmetic  form.  Depend  upon  it,  the  uni- 
verse will  never  really  be  understood  unless  it  may 
be  sometime  resolved  into  an  ordered  multiplicity  and 
made  to  own  itself  an  everlasting  drama  of  the  calculus. 

Let  us,  then,  trust  the  arithmetic  instinct  as  funda- 
mental and,  for  instruments  of  thought  that  shall  not 


MATHEMATICAL   EMANCIPATIONS  103 

fail,  repair  at  once  to  the  domain  of  number.  Every 
one  who  may  reasonably  aspire  to  a  competent  knowl- 
edge of  the  subject  in  hand  is  more  or  less  familiar  with 
the  system  of  real  numbers,  composed  of  the  positive 
and  negative  integers  and  fractions,  such  irrational 
numbers  as  V?  and  TT  and  countless  hosts  of  similar 
numbers  similarly  definable.  He  may  know  that,  for 
reasons  which  need  not  be  given  here,  the  system  of 
real  numbers  is  commonly  described  as  the  analytical 
continuum  of  second  order.  He  knows,  too,  at  any  rate 
it  is  a  fact  which  he  will  assume  and  readily  appreciate, 
that  the  distance  between  any  two  points  of  a  right  line 
is  exactly  expressible  by  a  number  of  the  continuum; 
that,  conversely,  given  any  number,  two  points  may  be 
found  whose  distance  apart  is  expressed  by  the  numerical 
value  of  that  number;  that,  therefore,  it  is  possible  to 
establish  a  unique  and  reciprocal,  or  one-to-one,  corre- 
spondence between  the  real  numbers  and  the  points 
of  a  straight  line,  namely,  by  assuming  some  point  of 
the  line  as  a  fixed  point  of  reference  or  origin  of  dis- 
tances, by  agreeing  that  a  distance  shall  be  positive  or 
negative  according  as  it  proceeds  from  the  origin  in 
this  sense  or  in  the  other  and  by  agreeing  that  a  point 
and  the  number  which  by  its  magnitude  reckoned  in 
terms  of  a  chosen  finite  unit  however  great  or  small 
serves  to  express  the  distance  of  the  point  from  the 
origin  and  by  its  sign  indicates  on  which  side  of  the 
origin  the  point  is  situated,  shall  be  a  pair  of  corre- 
spondents. Accordingly,  if  the  point  P  glides  along 
the  line,  the  corresponding  number  v  will  vary  in  such 
a  way  that  to  each  position  of  the  geometric  there 
corresponds  one  value  of  the  arithmetic  element,  and 
conversely.  P  represents  v;  and  v,  P.  No  two  P's 
represent  a  same  p;  and  no  two  c's,  a  same  P.  By 


104  MATHEMATICAL  EMANCIPATIONS 

virtue  of  the  correlation  thus  established  with  the  analyt- 
ical continuum,  we  may  describe  the  line  as  a  simple  or 
one-fold  geometric  continuum,  namely,  of  points.  The 
like  may  in  general  be  said,  and  for  the  same  reason, 
of  any  curve  whatever,  but  we  select  the  straight  line 
as  being  the  simplest,  for  in  matters  fundamental  we 
should  prefer  clearness  to  riches  of  illustration,  in  the 
faith  that,  if  first  we  seek  the  former,  the  latter  shall 
in  due  course  be  added  unto  it.  The  straight  line, 
when  it  is  regarded  as  the  domain  of  geometric  opera- 
tion, as  the  region  or  room  containing  the  configurations 
or  sets  of  elements  with  which  we  deal,  is  and  is  called 
a  space;  and  this  space,  viewed  as  the  manifold  or 
assemblage  of  its  points,  is  said  to  be  0w£-dimensional 
for  the  reason  that,  as  we  have  seen,  in  order  to  deter- 
mine the  position  of  a  point  in  it,  in  order,  i.e.,  to  pick 
out  or  distinguish  a  point  from  all  the  other  points  of 
the  manifold,  it  is  necessary  and  sufficient  to  know  one 
fact  about  the  point,  as,  e.g.,  its  distance  from  an  as- 
sumed point  of  reference.  In  other  words,  the  line 
is  called  a  one-dimensional  space  of  points  because  in 
that  space  the  point  has  one  and  but  one  degree  of 
freedom  or,  what  is  tantamount,  because  the  position 
of  the  point  depends  upon  the  value  of  a  single  v, 
known  as  its  coordinate. 

Herewith  is  immediately  suggested  the  generic  con- 
cept of  dimensionality:  if  an  assemblage  of  elements  of 
any  given  kind  whatsoever,  geometric  or  analytic  or  neither, 
as  points,  lines,  circles,  triangles,  numbers,  notions,  senti- 
ments, hues,  tones,  be  such  that,  in  order  to  distinguish 
every  element  of  the  assemblage  from  all  the  others,  it  is 
necessary  and  sufficient  to  know  exactly  n  independent  facts 
about  the  element,  then  the  assemblage  is  said  to  be  n- 
dimensional  in  the  elements  of  the  given  kind.  It  appears, 


MATHEMATICAL  EMANCIPATIONS  105 

therefore,  that  the  notion  of  dimensionality  is  by  no 
means  exclusively  associated  with  that  of  space  but  on 
the  contrary  may  often  be  attached  to  the  far  more 
generic  concept  of  assemblage,  aggregate  or  manifold. 
For  example,  duration,  the  total  aggregate  of  time-points, 
or  instants,  is  a  simple  or  one-fold  assemblage.  On  the 
other  hand,  the  assemblage  of  colors  is  three-dimensional 
as  is  also  that  of  musical  notes,  for  in  the  former  case, 
as  shown  by  Clerk  Maxwell,  Thomas  Young  and  others, 
every  color  is  composable  as  a  definite  mixture  of  three 
primary  ones  and  so  depends  upon  three  independent 
variables  or  coordinates  expressing  the  amounts  of  the 
fundamental  components.  And  in  the  latter  case  a 
similar  scheme  obtains,  one  note  being  distinguishable 
from  all  others  when  and  only  when  the  three  general 
marks,  pitch,  length,  and  loudness,  are  each  of  them 
specified.  In  passing  it  seems  worth  while  to  point  out 
the  possibility  of  appropriating  the  name  soul  to  signify 
the  manifold  of  all  possible  psychic  experiences,  in  which 
event  the  term  would  signify  an  assemblage  of  probably 
infinite  dimensionality,  and  the  assemblage  would  be 
continuous,  too,  if  Oswald  !  be  right  in  his  contention 
that  every  manifold  of  experience  possesses  the  character 
of  continuity.  That  contention,  however,  if  the  much 
abused  term  continuity  be  allowed  to  have  its  single 
precise  definitely  seizable  scientific  meaning,  is  far  less 
easy  to  make  good  than  that  eminent  chemist  and 
courageous  philosopher  seems  to  think. 

Returning  to  the  concept  of  space,  an  n-fold  assem- 
blage will  be  an  n-dimensional  space  if  the  elements  of 
the  assemblage  are  geometric  entities  of  any  given  kind. 
We  have  seen  that  the  straight  line  is  a  0ni-dimensional 
space  of  points.  But  in  studying  the  right  line  conceived 
>  Cf.  his  Natw-PkUotopkU. 


106  MATHEMATICAL  EMANCIPATIONS 

as  a  space,  we  are  not  compelled  to  employ  the  point 
as  element.  Instead  we  may  choose  to  assume  as  ele- 
ment the  point  pair  or  triplet  or  quatrain,  and  so  on.  The 
line  would  then  be  for  our  thought  primarily  a  space, 
not  of  points,  but  of  point  pairs  or  triplets  and  so  on, 
and  it  would  accordingly  be  strictly  a  space  of  two 
dimensions  or  of  three,  and  so  on;  for,  obviously,  to 
distinguish  say  a  point  pair  from  all  other  such  pairs 
we  should  have  to  know  two  independent  facts  about 
the  pair.  The  pair  would  have  two  degrees  of  freedom 
in  the  line,  its  determination  would  depend  upon  two 
independent  variables  as  v\  and  DZ-  These  variables 
might  be  the  two  independent  ratios  of  the  coefficients 
and  absolute  term  in  a  quadratic  equation  in  one  un- 
known, as  x,  for  to  know  the  ratios  is  to  know  the  equa- 
tion and  therewith  its  two  roots,  the  two  values  of  x. 
These  being  laid  off  on  the  line  give  the  point  pair. 
Conversely,  a  point  pair  gives  two  values  of  x,  hence 
definite  quadratic  equation  and  so  values  of  v\  and  %. 
On  its  arithmetic  side  the  shield  presents  a  precisely 
parallel  doctrine.  The  simple  analytical  continuum 
composed  of  the  real  numbers  immediately  loses  its 
simplicity  and  assumes  the  character  of  a  2-  or  3-  ... 
or  w-fold  analytical  continuum  if,  instead  of  thinking  of 
its  individual  numbers,  we  view  it  as  an  aggregate  of 
number  pairs  or  triplets  or,  in  general,  as  the  totality 
of  ordered  systems  of  n  numbers  each. 

In  the  light  of  the  preceding  paragraph  it  is  seen  that 
the  dimensionality  of  a  given  space  is  not  unique  but 
depends  upon  the  choice  of  geometric  entity  for  primary 
or  generating  element.  A  space  being  given,  its  dimen- 
sionality is  not  therewith  determined  but  depends  upon 
the  will  of  the  investigator,  who  by  a  proper  choice 
of  generating  elements  may  endow  the  space  with  any 


MATHEMATICAL  EMANCIPATIONS  107 

dimensionality  he  pleases.  That  fact  is  of  cardinal 
significance  alike  for  science  and  for  philosophy.  I 
reserve  for  a  little  while  its  further  consideration  in  order 
to  present  at  once  a  kind  of  complementary  fact  of  equal 
interest  and  of  scarcely  less  importance.  It  is  that  two 
spaces  which  in  every  other  respect  are  essentially  un- 
like, thoroughly  disparate,  may,  by  suitable  choice  of 
generating  elements,  be  made  to  assume  equal  dimen- 
sionalities. Consider,  for  example,  the  totality  of  lines 
contained  in  a  same  plane  and  containing  a  point  in 
common.  Such  a  totality,  called  a  pencil,  of  lines  is  a 
simple  geometric  continuum,  namely,  of  lines.  It  is, 
then,  and  may  be  called,  a  0n«-dimensional  space  of 
lines  just  as  the  line  or  range  of  points  is  a  one-dimen- 
sional space  of  points.  The  two  spaces  are  equally 
rich  in  their  respective  elements.  And  if,  following 
Desargues  and  his  successors,  we  adjoin  to  the  points 
of  the  range  a  so-called  "ideal"  point  or  point  at  infinity, 
thus  rendering  the  range  like  the  pencil,  closed,  it  be- 
comes obvious  that  two  intelligences,  adapted  and 
confined  respectively  to  the  two  simple  spaces  in  ques- 
tion, would  enjoy  equal  freedom;  their  analytical  experi- 
ences would  be  identical,  and  their  geometries,  though 
absolutely  disparate  in  kind,  would  be  equally  rich  in 
content.  Just  as  the  range-dweller  would  discover  that 
the  dimensionality  of  his  space  is  two  in  point  pairs, 
three  in  triplets,  and  so  on,  so  the  pencil-inhabitant  would 
find  his  space  to  be  of  dimensionality  two  in  line  pairs, 
three  in  triplets,  and  so  on  without  end.  It  was  indi- 
cated above  that  any  line,  straight  or  curved,  is  a  one- 
dimensional  space  of  points.  In  that  connection  it 
remains  to  say  that,  speaking  generally,  any  curve, 
literally  and  strictly  conceived  as  the  assemblage  of  its 
(tangent)  lines  and  so  including  the  point  or  pencil 


108  MATHEMATICAL  EMANCIPATIONS 

as  a  special  case,  is  also  a  0w0-dimensional  space  of  lines. 
It  is,  moreover,  obvious  that  the  foregoing  considera- 
tions respecting  the  range  of  points  and  the  pencil  of 
lines  are,  mutatis  mutandis,  equally  valid  for  any  one 
of  an  infinite  variety  of  other  analogous  spaces,  as,  e.  g., 
the  axal  pencil,  a  one-fold  space  of  planes,  consisting 
of  the  totality  of  planes  having  a  line  in  common. 

If  perchance  some  reader  should  feel  an  ungrateful 
sense  of  impropriety  in  our  use  of  the  term  space  to 
signify  such  common  geometric  aggregates  as  we  have 
been  considering,  I  gladly  own  that  his  state  of  mind 
is  a  perfectly  natural  one.  But  it  is,  besides  and  on 
that  account,  a  source  of  real  encouragement.  Dictional 
sensibility  is  a  hopeful  sign,  being  conclusive  evidence 
of  life,  and,  while  there  is  life,  there  remains  the  pos- 
sibility and  therewith  the  hope  of  readjustment.  In 
the  present  case,  £  venture  to  assure  the  reader,  on 
grounds  both  of  personal  experience  and  of  the  experi- 
ence of  others,  that  whatever  sense  he  may  have  of 
injury  received  will  speedily  disappear  in  the  further 
course  of  his  meditations.  Only,  let  him  not  be  im- 
patient. Larger  meanings  must  have  time  to  grow; 
the  smaller  ones,  those  that  are  most  natural  and  most 
provincial,  being  also  the  most  persistent.  In  the 
process  of  clarification,  expansion  and  readjustment, 
his  fine  old  word,  space,  early  come  into  his  life  and 
gradually  stained  through  and  through  with  the  re- 
fracted partial  lights  and  multi-colored  prejudices  of 
his  youth,  is  not  to  be  robbed  of  its  proper  charms  nor 
to  be  shorn  of  its  proper  significance.  More  than  it  will 
lose  of  mystery,  it  shall  gain  of  meaning.  Of  this  last 
it  has  hitherto  had  for  him  but  little  that  was  of  sci- 
entific value,  but  little  that  was  not  vague  and  elusive 
and  ultimately  unseizable.  That  was  because  the  word 


MATHEMATICAL  EMANCIPATIONS  IOQ 

stood  for  something  absolutely  sui  generis,  i.  e.,  for  a 
genus  neither  including  species  nor  being  itself  included 
in  a  class.  But  now,  on  the  other  hand,  both  of  these 
negatives  are  henceforth  to  be  denied,  and  the  hitherto 
baffling  term,  perfect  symbol  of  the  unthinkable,  always 
promising  and  never  presenting  definable  content, 
immediately  assumes  the  characteristic  twofold  aspect 
of  a  genuine  concept,  being  at  once  included  as  member 
of  a  higher  class,  the  more  generic  class  of  manifolds, 
and  including  within  itself  an  endless  variety  of  indi- 
viduals, an  infinitude  of  species  of  space. 

Of  these  species,  the  next  in  order  of  simplicity,  to 
those  above  considered,  is  the  plane.  To  distinguish  a 
point  of  a  plane  from  all  its  other  points,  it  is  necessary 
and  sufficient  to  know  two  independent  facts  about  its 
position,  as,  e.  g.,  its  distances  from  two  assumed  lines 
of  reference,  most  conveniently  taken  at  right  angles. 
Viewed  as  the  ensemble  of  its  points,  the  plane  is,  there- 
fore, a  space  of  two  dimensions.  In  that  space,  the 
point  enjoys  a  freedom  exactly  twice  that  of  a  point 
in  a  range  or  of  a  line  in  a  pencil,  and  exactly  equal  to 
that  of  a  pair  of  points  or  of  lines  in  the  last-mentioned 
spaces.  On  the  other  hand,  if  the  point  pair  be  taken 
as  element  of  the  plane,  the  latter  becomes  a  space  of 
four  dimensions. 

What  if  the  line  be  taken  as  generating  element  of  the 
plane?  It  is  obvious  that  the  plane  is  equally  rich  in 
pencils  and  in  ranges.  It  contains  as  many  lines  as 
points,  neither  more  nor  less.  Two  points  determine  a 
line;  two  lines,  a  point;  if  the  lines  be  parallel,  their 
common  point  is  a  Desarguesian,  a  point  at  infinity. 
We  should  therefore  expect  to  find  that  in  a  plane  the 
position  of  a  line  depends  upon  two  and  but  two  inde- 
pendent variables.  And  the  expectation  is  realized,  as 


110  MATHEMATICAL  EMANCIPATIONS 

it  is  easy  to  see.  For  if  the  variables  be  taken  to  repre- 
sent (say)  distances  measured  from  chosen  points  along 
two  lines  of  reference,  it  is  immediately  evident  that  a 
given  pair  of  values  of  the  variables  determines  a  line 
uniquely  and  that,  conversely,  a  given  line  uniquely 
determines  such  a  pair.  The  plane  is,  therefore,  a  too- 
dimensional  space  of  lines  as  well  as  of  points.  In  line 
pairs,  as  in  point  pairs,  its  dimensionality  is  four.  We 
may  suppose  the  space  in  question  to  be  inhabited  by 
two  sorts  of  individuals,  one  of  them  capable  of  thinking 
in  terms  of  points  but  not  of  lines,  the  other  in  terms  of 
lines  but  not  of  points.  Each  would  find  his  space 
bi-dimensional.  They  would  enjoy  precisely  the  same 
analytical  experience.  Between  their  geometries  there 
would  subsist  a  fact-to-fact  correspondence  but  not  the 
slightest  resemblance.  For  example,  the  circle  would 
be  for  the  former  individual  a  certain  assemblage  of 
points  but  devoid  of  tangent  lines,  and,  for  the  latter, 
a  corresponding  assemblage  of  (tangent)  lines  but  devoid 
of  contact  points. 

Passing  from  the  plane  to  a  curved  surface,  to  a 
sphere,  for  example,  a  little  reflection  suffices  to  show 
that  the  latter  may  be  conceived  in  a  thousand  and  one 
ways,  but  most  simply  as  the  ensemble  of  its  points  or 
of  its  (tangent)  planes  or  of  its  (tangent)  lines.  These 
various  concepts  are  logically  equivalent  and  in  them- 
selves are  equally  intelligible.  And  if  to  us  they  do 
not  seem  to  be  also  equally  good,  that  is  doubtless  be- 
cause we  are  but  little  traveled  in  the  great  domain  of 
Reason  and  therefore  naturally  prefer  our  familiar 
customs  and  provincial  points  of  view  to  others  that  are 
strange.  At  all  events,  it  is  certain  that  on  purely 
rational  grounds,  none  of  the  concepts  in  question  is  to 
be  preferred,  while,  from  preference  based  on  other 


MATHEMATICAL  EMANCIPATIONS  III 

grounds,  it  is  the  office  alike  of  science  and  of  philosophy 
to  provide  the  means  of  emancipation.  Let  us,  then, 
detach  ourselves  from  the  vulgar  point  of  view  and  for 
a  moment  contemplate  the  three  concepts  as  coordi- 
nate indeed  but  independent  concepts  of  surface.  And 
for  the  sake  of  simplicity,  we  may  think  of  a  sphere.1 
Suppose  it  placed  upon  a  plane  and  imagine  its  highest 
point,  which  we  may  call  the  pole,  joined  by  straight 
lines  to  all  the  points  of  the  plane.  Each  line  pierces 
the  sphere  in  a  second  point.  It  is  plain  that  thus  a 
one-to-one  correspondence  is  set  up  between  the  points 
of  the  sphere  and  those  of  the  plane,  except  that  the 
pole  corresponds  at  once  to  all  the  Desarguesian  points 
of  the  plane  —  an  exception,  however,  which  is  here  of 
no  importance.  The  plane  and  the  sphere  are,  then, 
equally  rich  in  points.  Accordingly,  the  sphere  con- 
ceived as  a  plenum  or  locus  or  space  of  points  is  a  space 
of  two  dimensions.  In  that  space  the  point  has  two 
degrees  of  freedom.  Its  position  depends  upon  two 
independent  variables,  as  latitude  and  longitude.  But 
we  may  conceive  the  surface  quite  otherwise:  at  each 
of  its  points  there  is  a  (tangent)  plane,  and  now,  dis- 
regarding points,  we  may  think  only  of  the  assemblage 
of  those  planes.  These  together  constitute  a  sphere, 
not,  however,  as  a  locus  of  points,  but  as  an  envelope 
(as  it  is  called)  of  planes.  And  what  shall  we  say  of 
the  surface  as  thus  conceived?  The  answer  obviously 
is  that  it  is  a  taw-dimensional  space  of  planes,  admitting 
of  a  geometry  quite  as  rich  and  as  definite  as  is  the 
theory  of  any  other  space  of  equal  dimensionality.  In 
each  of  the  planes  there  is  a  pencil  of  lines  of  which  each 
is  tangent  to  the  sphere.  Thus  we  are  led  to  a  third 

1  The  term  is  here  employed  as  in  the  higher  geometry  to  denote,  not  a 
solid,  byt  a  surface. 


112  MATHEMATICAL  EMANCIPATIONS 

conception  of  our  surface.  We  have  merely  to  dis- 
regard both  points  and  planes  and  confine  our  atten- 
tion to  the  assemblage  of  lines.  The  vision  which  thus 
arises  is  that  of  a  /Aree-dimensional  space  of  lines. 
In  pencils,  its  dimensionality  is  two.  In  this  space  the 
pencil  has  two  and  the  line  three  degrees  of  freedom. 

But  let  us  return  to  the  plane.  We  have  seen  that  at 
the  geometrician's  bidding  it  plays  the  r61e  of  a  two- 
fold space  either  in  points  or  in  lines.  It  is  natural  to 
ask  whether  it  may  be  conceived  as  a  space  of  three 
dimensions,  like  the  sphere  in  its  third  conception.  The 
answer  is  affirmative:  it  may  be  so  conceived,  and  that 
in  an  infinity  of  ways.  Of  these  the  simplest  is  to  as- 
sume the  circle  as  primary  or  generating  element.  Of 
circles  the  plane  contains  a  threefold  infinity,  an  infinity 
of  infinities  of  infinities.  It  is  a  circle  continuum  of 
third  order.  To  distinguish  any  one  of  its  circles  from 
all  the  rest,  three  independent  data,  two  for  position  and 
one  for  size,  are  necessary  and  sufficient.  In  the  plane 
the  circle  has  three  degrees  of  freedom,  its  determination 
depends  upon  three  independent  variables.  The  plane 
is,  accordingly,  a  tri:dimensional  space  of  circles.  In 
parabolas  its  dimensionality  is  four;  in  conies,  five;  and 
so  on  without  limit. 

Before  turning  to  space,  ordinarily  so-called,  it  seems 
worth  while  to  indicate  another  geometric  continuum 
which,  although  it  presents  no  likeness  whatever  to 
the  plane,  nevertheless  matches  it  perfectly  in  every  con- 
ceptual aspect.  The  reference  is  to  the  sheaf,  or  bundle, 
of  lines,  i.  e.,  the  totality  of  lines  having  a  point  in 
common.  The  point  is  to  be  disregarded  and  the  lines 
viewed  as  non-decomposable  entities,  like  points  in  a  line 
or  plane  regarded  as  an  assemblage  of  points.  Thus 
conceived,  the  sheaf  is  literally  a  space,  namely,  of  lines. 


MATHEMATICAL  EMANCIPATIONS  113 

It  is,  in  the  vulgar  sense  of  the  term,  just  as  big,  occu- 
pies precisely  as  much  room,  nay  indeed  the  same  room, 
as  the  space  in  which  we  live.  The  sheaf  as  a  space  is 
taw-dimensional  in  lines,  like  the  plane  in  points;  two- 
dimensional  in  pencils,  like  the  plane  in  lines;  four- 
dimensional  in  line  or  pencil  pairs,  like  the  plane  in 
point  or  line  pairs;  /Aree-dimensional  in  ordinary  cones, 
like  the  plane  in  circles;  and  so  on  and  on. 

In  the  light  of  the  foregoing  considerations,  any  hith- 
erto uninitiated  reader  will  probably  suspect  that  ordi- 
nary space  is  not,  as  it  is  commonly  supposed  and  said 
to  be,  an  inherently  and  uniquely  /Arce-dimensional 
affair.  His  suspicion  is  completely  justified  by  fact. 
The  simple  traditional  affirmation  of  tri-dimensionality 
is  devoid  of  definite  meaning.  It  is  unconsciously  elliptic, 
requiring  for  its  completion  and  precision  the  specifica- 
tion of  an  appropriate  geometric  entity  for  generating 
element.  Merely  to  say  that  space  is  tri-dimensional 
because  a  solid,  e.  g.,  a  plank,  has  length,  breadth  and 
thickness,  is  too  crude  for  scientific  purposes.  More- 
over, it  betrays,  quite  unwittingly  indeed  as  we  shall 
see,  an  exceedingly  meager  point  of  view.  Not  only 
does  it  assume  the  point  as  element  but  it  does  so  tacitly 
because  unconsciously,  as  if  the  point  were  not  merely 
an  but  the  element  of  ordinary  space.  An  element  the 
point  may  obviously  be  taken  to  be,  and  in  that  ele- 
ment ordinary  space  is  indeed  tri-dimensional,  for  the 
position  of  a  point  at  once  determines  and  is  determined 
by  three  independent  data,  as  its  distances  from  three 
assumed  mutually  perpendicular  planes  of  reference. 
It  must  be  admitted,  too,  that  the  point  does,  in  a  sense, 
recommend  itself  as  the  element  par  excellence,  at  least 
for  practical  purposes.  For  example,  we  prefer  to  do  our 
drawing  with  the  point  of  a  pencil  to  doing  it  with  a 


114  MATHEMATICAL  EMANCIPATIONS 

straight  edge.  But  that  is  a  matter  of  physical  as  dis- 
tinguished from  rational  convenience.  Preference  for 
the  point  has,  then,  a  cause:  in  the  order  of  evolution, 
practical  man  precedes  man  rational  and  determines 
for  the  latter  his  initial  choices.  Causes,  however,  are 
extra-logical  things,  and  the  preference  in  question, 
though  it  has  indeed  a  cause,  has  no  reason.  Accord- 
ingly, when  in  these  modern  times,  the  geometrician 
became  clearly  conscious  that  he  was  in  fact  and  had 
been  from  time  immemorial  employing  the  point  as 
element  and  that  it  was  this  use  that  lent  to  space 
its  traditional  triplicity  of  dimensions,  he  did  not  fail 
to  perceive  almost  immediately  the  logically  equal 
possibility  of  adopting  at  will  for  primary  element  any 
one  of  an  infinite  variety  of  other  geometric  entities  and 
so  the  possibility  of  rationally  endowing  ordinary  space 
with  any  prescribed  dimensionality  whatever. 

Thus,  for  example,  the  plane  is  no  less  available 
for  generating  element  than  is  the  point.  The  plane 
is  logically  and  intuitionally  just  as  simple,  for,  if  from 
force  of  habit,  we  are  tempted  to  analyze  the  plane 
into  an  assemblage  of  points,  the  point  is  in  its  turn 
equally  conceivable  as  or  analyzable  into  an  assemblage 
of  planes,  the  sheaf  of  planes  containing  the  point. 
We  may,  then,  regard  our  space  as  primarily  a  plenum 
of  planes.  To  determine  a  plane  requires  three  and 
but  three  independent  data,  as,  say,  the  distances  to 
it  measured  along  three  chosen  lines  from  chosen  points 
upon  them.  It  follows  that  ordinary  space  is  three- 
dimensional  in  planes  as  well  as  in  points.  But  now 
if  (with  Pliicker)  we  think  of  the  line  as  element,  we 
shall  find  that  our  space  has  four  dimensions.  That  fact 
may  be  seen  in  various  ways,  most  easily  perhaps  as 
follows.  A  line  is  determined  by  any  two  of  its  points. 


MATHEMATICAL   EMANCIPATIONS  11$ 

Every  line  pierces  every  plane.  By  joining  the  points 
of  one  plane  to  all  the  points  of  another,  all  the  lines 
of  space  are  obtained.  To  determine  a  line  it  is,  then, 
enough  to  determine  two  of  its  points,  one  in  the  one 
plane  and  one  in  the  other.  For  each  of  these  deter- 
minations, two  data,  as  before  explained,  are  necessary 
and  sufficient.  The  position  of  the  line  is  thus  seen  to 
depend  upon  four  independent  variables,  and  the  four- 
dimensionality  of  our  space  in  lines  is  obvious.  Again, 
we  may  (with  Lie)  view  our  space  as  an  assemblage  of 
its  spheres.  To  distinguish  a  sphere  from  all  other 
spheres,  we  need  to  know  four  and  but  four  independent 
facts  about  it,  as,  say,  three  that  shall  determine  its 
center  and  one  its  size.  Hence  our  space  is  four- 
dimensional  also  in  spheres.  In  circles  its  dimensionality 
is  six;  in  surfaces  of  second  order  (those  that  are  pierced 
by  a  straight  line  in  two  points),  nine;  and  so  on  ad 
infinitum. 

Doubtless  the  reader  is  prepared  to  say  that,  if  the 
foregoing  account  of  hyperspace  be  correct,  the  notion 
is  after  all  a  very  simple  one.  Let  him  be  assured,  the 
account  is  correct  and  his  judgment  is  just:  the  notion 
is  simple.  That  property,  as  said  in  the  beginning,  is 
indeed  one  of  its  merits.  As  presented  the  concept  is 
entirely  free  from  mystery.  To  seize  upon  it,  it  is  un- 
necessary to  pass  the  bounds  of  the  visible  universe  or 
to  transcend  the  limits  of  intuition.  Its  realization  is 
found  even  in  the  line,  in  the  pencil,  in  the  plane,  in  the 
sheaf,  here,  there  and  yonder,  everywhere,  in  fact.  The 
account,  however,  though  quite  correct,  is  not  yet  com- 
plete. The  term  hyperspace  has  yet  another  meaning 
and  yet  in  strictness  not  another,  as  we  shall  see.  It 
will  be  noticed  that  among  the  foregoing  examples  of 
hyperepace,  none  is  presented  of  dimensionality  exceed- 


Il6  MATHEMATICAL   EMANCIPATIONS 

ing  three  in  points.  It  is  precisely  this  variety  of  hyper- 
space  that  the  term  is  commonly  employed  to  signify, 
particularly  in  popular  enquiry  and  philosophical  specu- 
lation. And  it  is  this  variety,  too,  that  just  because  it 
baffles  the  ordinary  visual  imagination,  proves  to  be,  for 
the  non-mathematician  at  any  rate,  at  once  so  tanta- 
lizing, so  mysterious  and  so  fascinating. 

It  remains,  then,  to  ask,  what  is  meant  by  a  hyper- 
space  of  points?  How  is  the  notion  formed  and  what 
is  its  motivity  and  use?  The  path  of  enquiry  is  a  fa- 
miliar one  and  is  free  from  logical  difficulty.  Granted 
that  a  one-to-one  correspondence  can  be  established 
between  the  real  numbers  and  the  points  of  a  right  line, 
so  that  the  geometric  serve  to  represent  the  arithmetic 
elements;  granted  that  all  (ordered)  pairs  of  numbers  are 
similarly  representable  by  the  points  of  a  plane,  and  all 
(ordered)  triplets  by  the  points  of  ordinary  space;  the 
suggestion  then  naturally  presents  itself  that,  whether 
there  really  is  or  not,  there  ought  to  be  a  space  whose 
points  would  serve  to  represent,  as  in  the  preceding 
cases,  all  ordered  systems  of  values  of  n  independent 
variables;  and  especially  to  an  analyst  with  a  strong 
geometric  predilection,  to  one  who  is  a  born  Vorstellender 
for  whom  analytic  abstractions  naturally  tend  to  take 
on  figure  and  assume  the  exterior  forms  of  sense,  that 
suggestion  comes  with  a  force  which  he  alone  perhaps 
can  fully  appreciate.  And  what  does  he  do?  Not  find- 
ing the  desiderated  hyperspace  present  to  his  vision  or 
intuition  or  visual  imagination,  he  posits  it,  or  if 
you  prefer,  he  creates  it,  in  thought.  The  concept  of 
hyperspace  of  points  is  thus  seen  to  be  off-spring  of 
Arithmetic  and  Geometry.  It  is  legitimate  fruit  of  the 
indissoluble  union  of  the  fundamental  sciences. 

Does  such  hyperspace  exist?     It  does  exist  genuinely. 


MATHEMATICAL   EMANCIPATIONS  117 

If  not  for  intuition,  it  exists  for  conception;  if  not  for 
imagination,  it  exists  for  thought;  if  not  for  sense,  it 
exists  for  reason;  if  not  for  matter,  it  exists  for  mind. 
These  if's  are  ifs  in  fact.  The  question  of  imaginability 
is  really  a  question.  We  shall  return  to  it  presently. 

The  concept  of  hyperspace  of  points  is  generable  in 
various  other  ways.  Of  all  ways  the  following  is  per- 
haps the  best  because  of  its  appeal  at  every  stage  to 
intuition.  Let  there  be  two  points  and  grant  that  these 
determine  a  line,  point-space  of  one  dimension.  Next 
posit  a  point  outside  of  this  line  and  suppose  it  joined 
by  lines  to  all  the  points  of  the  given  line.  The  points 
of  the  joining  lines  together  constitute  a  plane,  point- 
space  of  two  dimensions.  Next  posit  a  point  outside  of 
this  plane  and  suppose  it  joined  by  lines  to  all  the 
points  of  the  plane.  The  points  of  all  the  joining  lines 
together  constitute  an  ordinary  space,  point-space  of 
three  dimensions.  The  clue  being  now  familiar  to  our 
hand,  let  us  boldly  pursue  the  opened  course.  Let  us 
overleap  the  limits  of  common  imagination,  transcend 
ordinary  intuition  as  being  at  best  but  a  non-essential 
auxiliary,  and  in  thought  posit  an  extra  point  that,  for 
thought  at  all  events,  shall  be  outside  the  space  last 
generated.  Suppose  that  point  joined  by  lines  to  all 
the  points  of  the  given  space.  The  points  of  the  join- 
ing lines  together  constitute  a  point-space  of  four 
dimensions.  The  process  here  applied  is  perfectly  clear 
and  obviously  admits  of  endless  repetition. 

Moreover,  the  process  is  equally  available  for  gener- 
ating hyperspaces  of  other  elements  than  points.  For 
example,  let  there  be  two  intersecting  lines  and  grant 
that  these  determine  a  pencil,  line-space  of  one  dimen- 
sion. Next  posit  a  line  (through  the  vertex)  outside  of 
the  given  pencil  and  suppose  it  joined  by  pencils  to  all 


Il8  MATHEMATICAL  EMANCIPATIONS 

the  lines  of  the  given  pencil.  The  lines  of  the  joining 
pencils  together  constitute  a  sheaf,  line-space  of  two 
dimensions.  Next  posit  a  line  (through  the  vertex) 
outside  of  the  sheaf  and  suppose  it  joined  by  pencils 
to  all  the  lines  of  the  sheaf.  The  lines  of  the  joining 
pencils  constitute  a  hypersheaf,  line-space  of  three  dimen- 
sions. The  next  step  plainly  leads  to  a  line-space  of 
four  dimensions;  and  so  on  ad  infinitum. 

And  now  as  to  the  question  of  imaginability.  Is  it 
possible  to  intuit  configurations  in  a  hyperspace  of 
points?  Let  it  be  understood  at  the  outset  that  that 
is  not  in  any  sense  a  mathematical  question,  and  mathe- 
matics as  such  is  quite  indifferent  to  whatever  answer  it 
may  finally  receive.  Neither  is  the  question  primarily 
a  question  of  philosophy.  It  is  first  of  all  a  psychological 
question.  Mathematicians,  however,  and  philosophers 
are  also  men  and  they  may  claim  an  equal  interest  per- 
haps with  others  in  the  profounder  questions  concern- 
ing the  potentialities  of  our  common  humanity.  The 
question,  as  stated,  undoubtedly  admits  of  affirmative 
answer.  For  the  lower  spaces,  with  which  the  imagina- 
tion is  familiar,  exist-  in  the  higher,  as  the  line  in  the 
plane,  and  the  plane  in  ordinary  space.  But  that  is  not 
what  the  question  means.  It  means  to  ask  whether 
it  is  possible  to  imagine  hyper-configurations  of  points, 
i.e.,  point-configurations  that  are  not  wholly  contained 
in  a  point-space  (like  our  own)  of  three  dimensions. 
It  is  impossible  to  answer  with  absolute  confidence. 
One  reason  is  that  the  term  imagination  still  awaits 
precision  of  definition.  Undoubtedly  just  as  three- 
dimensional  figures  may  be  represented  in  a  plane,  so 
four-dimensional  figures  may  be  represented  in  space. 
That,  however,  is  hardly  what  is  meant  by  imagining 
them.  On  the  other  hand,  a  four-dimensional  figure 


MATHEMATICAL  EMANCIPATIONS 

may  be  rotated  and  translated  in  such  a  way  that  all 
of  its  parts  come  one  after  another  into  the  threefold  do- 
main of  the  ordinary  intuition.  Again,  the  structure  of  a 
fourfold  figure,  every  minutest  detail  of  its  anatomy,  can 
be  traced  out  by  analogy  with  its  three-dimensional  ana- 
logue. Now  in  such  processes,  repetition  yields  skill, 
and  so  they  come  ultimately  to  require  only  amounts 
of  energy  and  of  time  that  are  quite  inappreciable.  Such 
skill  once  attained,  the  parts  of  a  familiar  fourfold  con- 
figuration may  be  made  to  pass  before  the  eye  of  in- 
tuition in  such  swift  and  effortless  succession  that  the 
configuration  seems  present  as  a  whole  in  a  single  instant. 
If  the  process  and  result  are  not,  properly  speaking, 
fourfold  imagination  and  fourfold  image,  it  remains  for 
the  psychologist  to  indicate  what  is  lacking. 

Certainly  there  is  naught  of  absurdity  in  supposing 
that  under  suitable  stimulation  the  human  mind  may 
in  course  of  time  even  speedily  develop  a  spatial  in- 
tuition of  four  or  more  dimensions.  At  present,  as  the 
psychologists  inform  us  and  as  every  teacher  of  geometry 
discovers  independently,  the  spatial  imagination,  in 
case  of  very  many  persons,  comes  distinctly  short  of 
being  strictly  even  tri-dimensional.  On  the  contrary,  it 
is  flat.  It  is  not  every  one,  even  among  scholars,  that 
with  eyes  closed  can  readily  form  a  visual  image  of  the 
whole  of  a  simple  solid  like  a  sphere,  enveloping  it  com- 
pletely with  bent  beholding  rays  of  psychic  light.  In 
such  defect  of  imagination,  however,  there  is  nothing 
to  astonish.  In  the  first  place,  man  as  a  race  is  only  a 
child.  He  has  been  on  the  globe  but  a  little  while, 
long  indeed  compared  with  the  fleeting  evanescents  that 
constitute  the  most  of  common  life,  but  very  short,  the 
merest  fraction  of  a  second,  in  the  infinite  stretch  of 
time.  In  the  second  place,  circumstances  have  not,  in 


120  MATHEMATICAL   EMANCIPATIONS 

general,  favored  the  development  of  his  higher  poten- 
tialities. His  chief  occupation  has  been  the  destruction 
and  evasion  of  his  enemies,  contention  for  mere  exist- 
ence against  hostile  environment.  Painful  necessity, 
then,  has  been  the  mother  of  his  inventions.  That,  and 
not  the  vitalizing  joy  of  self-realization,  has  for  the 
most  part  determined  the  selection  of  the  fashion  of  his 
faculties.  But  it  would  be  foolish  to  believe  that  these 
have  assumed  their  final  form  or  attained  the  limits  of 
their  potential  development.  The  imperious  rule  of 
necessity  will  relax.  It  will  never  pass  quite  away  but 
it  will  relax.  It  is  relaxing.  It  has  relaxed  appreciably. 
The  intellect  of  man  will  be  correspondingly  quickened. 
More  and  more  will  joy  in  its  activity  determine  its 
modes  and  forms.  The  multi-dimensional  worlds  that 
man's  reason  has  already  created,  his  imagination  may 
yet  be  able  to  depict  and  illuminate. 

It  remains  to  ask,  finally,  what  purpose  the  concept 
of  hyperspace  subserves.  Reply,  partly  explicit  but 
chiefly  implicit,  is  not,  I  trust,  entirely  wanting  in  what 
has  been  already  said.  Motivity,  at  all  events,  and 
raison  d'etre  are  not  far  to  seek.  On  the  one  hand,  the 
great  generalization  has  made  it  possible  to  enrich, 
quicken  and  beautify  analysis  with  the  terse,  sensuous, 
artistic,  stimulating  language  of  geometry.  On  the 
other  hand,  the  hyperspaces  are  in  themselves  im- 
measurably interesting  and  inexhaustibly  rich  fields  of 
research.  Not  only  does  the  geometrician  find  light  in 
them  for  the  illumination  of  many  otherwise  dark  and 
undiscovered  properties  of  the  ordinary  spaces  of  in- 
tuition, but  he  also  discovers  there  wondrous  struc- 
tures quite  unknown  to  ordinary  space.  These  he 
examines.  He  handles  them  with  the  delicate  instru- 
ments of  his  analysis.  He  beholds  them  with  the  eye 


MATHEMATICAL  EMANCIPATIONS  121 

of  the  understanding  and  delights  in  the  presence  of 
their  supersensuous  beauty. 

Creation  of  hyperspaces  is  one  of  the  ways  by  which 
the  rational  spirit  secures  release  from  limitation.  In 
them  it  lives  ever  joyously,  sustained  by  an  unfailing 
sense  of  infinite  freedom. 


THE    UNIVERSE    AND    BEYOND:    THE 
EXISTENCE  OF  THE  HYPERCOSMIC l 

Ni  la  contradiction  n'est  marque  de  faussete,  ni  I'incontradiction  n'est 
marque  de  verite.  —  PASCAL 

THE  inductive  proof  of  the  doctrine  of  evolution 
seems  destined  to  be  ultimately  judged  as  the  great 
contribution  of  Natural  Science  to  modern  thought. 
Among  the  presuppositions  of  that  doctrine,  among  the 
axioms,  as  one  may  call  them,  of  science,  are  found  the 
following:  — 

(1)  The    assumption    of    the    universal    and    eternal 
reign   of  law:    the   assumption   that   the  universe,  the 
theatre  of  evolution,  the  field  of  natural  science,  is  and 
eternally    has    been    a    genuine    Cosmos,    an    incarnate 
rational   logos,    an   embodiment   of   reason,    an   organic 
affair  of  order,  a  closed  domain  of  invariant  uniformities, 
in  which  waywardness  and  chance  have  had  nor  part 
nor  lot:    an  infinitely  intricate  garment,  ever  changing, 
yet  always  essentially  the  same,  woven,  warp  and  weft 
alike,  of  mathetic  relationships. 

(2)  The  assumption,  not  merely  that  the  universe  is 
cosmic  through  and  through,  but  that  it  is  the  all  con- 
junctively —  the   all,    that   is,   in   the   sense   of   naught 
excluded;   the  assumption,  in  other  words,  that  it  is  not 
merely  a  but  the  cosmos,  the  sole  system  of  law  and 
order  and  harmony,  the  complete  and  perfect  embodi- 
ment of  the  whole  of  truth. 

1  Appeared  in  The  Hibbert  Journal,  January,  1905. 


THE   UNIVERSE   AND   BEYOND  1 23 

Such,  I  take  it,  are  among  the  principles,  the  articles 
of  faith,  more  or  less  consciously  held  by  the  great 
majority  of  the  men  of  science  and  their  adherents. 

As  for  myself,  I  am  unable  to  hold  these  tenets  either 
as  self-evident  truths,  or  as  established  facts,  or  as  prop- 
ositions the  proof  of  which  may  be  confidently  awaited. 
Truth,  for  example,  especially  when  contemplated  in  its 
relations  to  curiosity  —  at  once  the  psychic  product  and 
psychic  agency  of  evolution  —  less  seems  a  completed 
thing  coeval  with  the  world  than  a  thing  derived  and 
still  becoming.  Again,  while  the  assumption  of  the 
cosmic  character  of  our  universe  is  of  the  greatest  value 
as  a  working  hypothesis,  I  am  unable  to  find  in  the 
method  of  natural  science  or  in  that  of  mathematics 
any  ground,  even  the  slightest,  for  expecting  conclusive 
proof  of  its  validity.  In  striking  contrast,  on  the  other 
hand,  with  this  negative  thesis,  there  is  found  in  the 
realm  of  pure  thought,  in  the  domain  of  mathematics, 
very  convincing  evidence,  not  to  say  indubitable  proof 
of  the  proposition,  that  no  single  cosmos,  whether  our 
universe  be  such  or  not,  can  enclose  every  rationally 
constructive  system  of  truth,  but  that  any  universe  is 
a  component  of  an  extra-universal,  that  above  every 
nature  is  a  super-natural,  beyond  every  cosmos  a 
hypercosmic. 

These  are  among  the  theses  presented  in  the  following 
pages,  not  in  a  controversial  spirit,  let  me  add,  nor 
accompanied  by  the  minuter  arguments  upon  which 
they  ultimately  rest. 

We  all  must  allow  that  truth  is.  To  deny  it  denies 
the  denial.  Such  scepticism  is  cut  away  by  the  sweep- 
ing blade  of  its  own  unsparing  doubt.  But  what  it  is  — 
that  is  another  matter.  The  assumption  that  truth  is 
an  agreement  or  correspondence  between  concepts  and 


124  THE   UNIVERSE   AND   BEYOND 

things,  between  thought  and  object,  is  of  very  great 
value  in  practical  affairs;  it  very  well  serves,  too,  the 
immediate  purposes  of  natural  science,  especially  in  its 
cruder  stage,  before  it  has  learned  by  critical  reflection 
on  its  own  processes  and  foundations  to  suspect  its 
limitations,  and  while,  like  the  proverbial  "chesty" 
youth  who  disdains  the  meagre  wisdom  of  his  father,  it  is 
apt  to  proclaim,  innocently  enough  if  somewhat  boast- 
fully, a  lofty  contempt  for  all  philosophy  and  meta- 
physics. Although  the  assumption  has  the  undoubted 
merit  of  being  thus  useful  in  high  degree,  it  is,  when 
regarded  as  a  definitive  formulation  of  what  we  mean 
by  truth,  hardly  to  be  accepted.  For,  not  only  does  it 
imply  —  what  may  indeed  be  quite  correct,  but  is  far 
from  being  demonstrated,  and  far  from  being  uni- 
versally allowed  —  namely,  that  "thing"  is  one  and 
"concept"  another,  that  "object"  and  "thought"  are 
twain,  but  even  if  we  grant  such  ultimate  implied 
duality,  it  remains  to  ask  what  that  "agreement"  is, 
or  "correspondence,"  that  mediates  the  hemispheres 
and  gives  the  whole  its  truth.  The  assumption  is 
slightly  too  na'ive  and"  unsophisticated,  a  little  too  redo- 
lent of  an  untamed  soil  and  primitive  stage  of  cultiva- 
tion. Much  profounder  is  that  insight  of  Hegel's,  that 
truth  is  the  harmony  which  prevails  among  the  objects 
of  thought.  If,  with  that  philosopher,  we  identify 
object  and  thought,  we  have  at  once  the  pleasing  utter- 
ance that  truth  is  the  harmony  of  ideas.  But  here, 
again,  easy  reflection  quickly  finds  no  lack  of  difficulties. 
For  what  should  we  say  an  idea  is?  And  is  there  really 
nothing  else,  except,  of  course,  their  harmony?  And 
what  is  that?  And  is  there  no  such  thing  as  contra- 
diction and  discord?  Is  that,  too,  a  kind  of  truth,  a 
kind  of  harmonious  jangling,  a  melody  of  dissonance? 


THE   UNIVERSE   AND   BEYOND  125 

The  fact  seems  to  be  that  truth  is  so  subtle,  diverse, 
and  manifold,  so  complex  of  structure  and  rich  in  as- 
pect, as  to  defy  all  attempt  at  final  definition.  Nay, 
more,  the  difficulty  lies  yet  deeper,  and  is  in  fact  irre- 
soluble.  Being  a  necessary  condition  thereto,  truth  can 
not  be  an  object  of  definition.  To  suppose  it  defined 
involves  a  contradiction,  for  the  definition,  being  some- 
thing new,  is  something  besides  the  truth  defined,  but 
it  must  itself  be  true,  and,  if  it  be,  in  that  has  failed  — 
the  enclosing  definition  is  not  itself  enclosed,  and 
straightway  asks  a  vaster  line  to  take  it  in,  and  so  ad 
in  fin  it  urn.  To  define  truth  would  be  to  construct  a 
formula  that  should  include  the  structure,  to  conceive 
a  water-compassed  ocean,  bounded  in  but  shutting 
nothing  out,  a  self-immersing  sea,  without  bottom  or 
surface  or  shore. 

Happily,  to  be  indefinable  is  not  to  be  unknowable 
and  not  to  be  unknown.  And  we  are  absolutely  certain 
that  truth,  whatever  it  may  be,  is  somehow  the  com- 
plement of  curiosity,  is  the  proper  stuff,  if  I  may  so 
express  it,  to  answer  questions  with.  Now  a  question, 
once  one  comes  to  think  of  it,  is  a  rather  odd  phenome- 
non. Half  the  secret  of  philosophy,  said  Leibnitz,  is 
to  treat  the  familiar  as  unfamiliar.  So  treated,  curi- 
osity itself  is  a  most  curious  thing.  How  blind  our 
familiar  assumptions  make  us!  Among  the  animals, 
man,  at  least,  has  long  been  wont  to  regard  himself  as 
a  being  quite  apart  from  and  not  as  part  of  the  cosmos 
round  about  him.  From  this  he  has  detached  himself 
in  thought,  he  has  estranged  and  objectified  the  world, 
and  lost  the  sense  that  he  is  of  it.  And  this  age-long 
habit  and  point  of  view,  which  has  fashioned  his  life 
and  controlled  his  thought,  lending  its  characteristic 
mark  and  colour  to  his  whole  philosophy  and  art  and 


126  THE   UNIVERSE  AND   BEYOND 

learning,  is  still  maintained,  partly  because  of  its  con- 
venience no  doubt,  and  partly  by  force  of  inertia  and 
sheer  conservatism,  in  the  very  teeth  of  the  strongest 
probabilities  of  biologic  science.  Probably  no  other  single 
hypothesis  has  less  to  recommend  it,  and  yet  no  other 
so  completely  dominates  the  human  mind.  Suppose 
we  deny  the  assumption,  as  we  seem  indeed  com- 
pelled to  do,  in  the  name  of  science,  and  readjoin  our- 
selves in  thought,  as  we  have  ever  been  joined  in  fact, 
to  this  universe  in  which  we  live  and  have  our  being; 
the  other  half  of  the  secret  of  philosophy  will  be  re- 
vealed, or  illustrated  at  all  events,  in  the  strangeness 
of  aspect  presented  by  things  before  familiar.  Note  the 
radical  character  of  the  transformation  to  be  effected. 
The  world  shall  no  longer  be  beheld  as  an  alien  thing, 
beheld  by  eyes  that  are  not  its  own.  Conception  of  the 
whole  and  by  the  whole  shall  embrace  us  as  part, 
really,  literally,  consciously,  as  the  latest  term,  it  may 
be,  of  an  advancing  sequence  of  developments,  as  occu- 
pying the  highest  rank  perhaps  in  the  ever-ascending 
hierarchy  of  being,  but,  at  all  events,  as  emerged  and 
still  emerging  natura  •  naturata  from  some  propensive 
source  within.  I  grant  that  the  change  in  point  of  view 
is  hard  to  make  —  old  habits,  like  walls  of  rock,  tend- 
ing to  confine  the  tides  of  consciousness  within  their 
accustomed  channels  —  but  it  can  be  made  and,  by 
assiduous  effort,  in  the  course  of  time,  maintained. 
Suppose  it  done.  By  that  reunion,  the  whole  regains, 
while  the  part  retains,  the  consciousness  the  latter  pur- 
loined. I  cannot  pause  to  note  even  the  most  striking 
consequences  of  such  a  change  in  point  of  view.  Time 
would  fail  me  to  follow  far  the  opening  lines  of  specu- 
lation that  issue  thence  and  invite  pursuit.  But  I 
cannot  refrain  from  pointing  out  how  exceedingly  curi- 


THE   UNIVERSE   AND   BEYOND  127 

ous  a  thing  curiosity  itself  becomes  when  beheld  and 
contemplated  from  the  mentioned  point  of  view.  For 
it  is  now  the  whole  that  meditates,  the  universe  that 
contemplates  —  a  once  mindless  universe  according  to 
its  present  understanding  of  the  term,  not  then  know- 
ing that  it  was,  unwittingly  unwitting  throughout  a 
beginningless  eternal  past  what  it  had  been  or  was  or 
was  to  be;  lawless,  too,  perhaps,  could  the  stream  of 
events  be  reascended,  though  blindly  and  slowly  be- 
coming lawful  through  habit- taking  tendence:  a  self- 
transforming  insensate  mass  composed  of  parts  without 
likeness  or  distinction,  continually  undergoing  change 
without  a  purpose,  devoid  of  passion,  and  neither  ig- 
norant nor  having  knowledge.  At  length  a  wondrous 
crisis  came,  an  event  momentous  —  when  or  how  is  yet 
unknown,  perhaps  through  fortuitous  concourse  of  part- 
less,  lawless,  wayward  elements.  At  all  events,  the  un- 
intending  tissues  formed  a  nerve,  the  universe  awoke 
alive  with  wonder,  mind  was  born  with  curiosity  and 
began  to  look  about  and  make  report  of  part  to  part 
and  thence  to  whole,  the  age  of  interrogation  was  at 
hand,  and  what  had  been  an  eternal  infinity  of  mindless 
being  began  to  question,  and  know  itself,  and  have  a 
sense  of  ignorance.  In  the  whole  universe  of  events, 
none  is  more  wonderful  than  the  birth  of  wonder,  none 
more  curious  than  the  nascence  of  curiosity  itself, 
nothing  to  compare  with  the  dawning  of  consciousness 
in  the  ancient  dark  and  the  gradual  extension  of  psychic 
life  and  illumination  throughout  a  cosmos  that  before 
had  only  been.  An  eternity  of  blindly  acting,  trans- 
forming, unconscious  existence,  assuming  at  length, 
through  the  birth  of  sense  and  intellect,  without  loss 
or  break  of  continuity,  the  abiding  form  of  fleeting 
time/  Another  eternity  remains  to  follow,  and  one 


128  THE  UNIVERSE   AND   BEYOND 

cannot  but  wonder  whether  there  shall  issue  forth  in 
future  from  the  marvel-weaving  loom  another  event,  or 
form  or  mode  of  being,  that  shall  be  to  the  modern 
universe  that  both  is  and  knows,  as  the  birth  of  soul  and 
curiosity  to  the  ancient  universe  that  was  but  did  not 
know.  A  speculation  by  no  means  idle,  but  let  it  pass. 

I  wish  to  point  out  next,  briefly,  that  curiosity  is  not 
only  a  principle  that  leads  to  knowing,  but  a  principle 
and  process  of  growing.  By  it  the  universe  comes  not 
merely  to  understand  itself,  but  actually  to  get  bigger 
thereby.  For  if  there  be  an  invariant  amount  of  matter, 
there  is  also  mind  increasing;  if  there  be  objects  that 
total  a  constant  sum,  there  are  also  ideas  that  multiply. 
A  new  query  and  a  new  answer  are  new  elements  in  the 
world,  by  which  the  latter  is  added  unto  and  enriched. 
Curiosity  is  the  aspect  of  the  universe  seeking  to  realise 
itself,  and  the  fruit  of  such  activity  is  new  reality,  stimu- 
lating to  new  research.  Imagine  a  body  with  an  inner 
core  of  outward-striving  impulses  producing  buds  at 
every  radial  terminus.  Such  is  knowledge  —  a  kind  of 
proliferating  sphere,  expanding  along  divergent  lines  by 
the  outward-seeking  of  an  inner  life  of  wonder.  Where- 
fore, it  appears  again  that  truth,  the  complement  of 
curiosity,  itself  grows  with  the  latter's  growth,  and, 
being  never  a  finished  thing,  but  one  that  both  is  and 
is  becoming,  is  not  to  be  compassed  by  definition  nor 
fully  solved  in  knowledge. 

In  respect  to  truth,  then,  the  upshot  is:  we  are  certain 
that  it  is;  not,  however,  as  a  closed  or  completed 
scheme  of  relationships,  but  as  a  kind  of  reality  charac- 
terised by  the  phenomena  of  growth  and  of  becoming; 
it  does  not  admit  of  ultimate  definition;  we  know,  how- 
ever, in  a  super-verbal  sense,  through  myriad  mani- 
festations of  it  to  a  faculty  in  us  of  feeling  for  it,  what 


THE   UNIVERSE  AND   BEYOND  I2Q 

it  is;  we  recognise  it  as  the  motive  power,  the  elixir 
vita,  the  sustaining  spring  of  wonder;  it  discovers  itself 
as  the  wherewithal  for  the  proper  fulfilment  of  the 
implicit  predictions  and  intimations  of  curiosity;  as  the 
thing  presaged  in  a  spiritual  craving,  confidently,  per- 
sistently proclaiming  its  needs  by  an  infinitude  of 
questionings. 

And  now  as  to  the  remainder  of  my  subject,  the  tale 
is  quite  too  long  to  be  told  in  full.  But  room  must 
be  found  for  a  partial  account,  for  important  fragments 
at  all  events. 

What,  then,  shall  we  say  mathematics  is?  A  question 
much  discussed  by  philosophers  and  mathematicians  in 
the  course  of  more  than  two  thousand  years,  and  espe- 
cially with  deepened  interest  and  insight  in  our  own 
time.  Many  an  answer  has  been  given  to  it,  but  none 
has  approved  itself  as  final.  Naturally  enough,  con- 
ception of  the  science  has  had  to  grow  with  the  science 
itself.  For  it  must  not  be  imagined  that  mathematics, 
because  it  is  so  old,  is  dead.  Old  it  is  indeed,  classic 
already  in  Euclid's  day,  being  surpassed  in  point  of 
antiquity  by  only  one  of  the  arts  and  by  none  of  the 
sciences;  but  it  is  also  living  and  new,  flourishing 
to-day  as  never  before,  advancing  in  a  thousand  direc- 
tions by  leaps  and  bounds.  It  is  not  merely  as  a  giant 
tree  throwing  out  and  aloft  myriad  branching  arms  in 
the  upper  regions  of  clearer  light,  and  plunging  deep 
and  deeper  roots  in  the  darker  soil  beneath.  It  is  rather 
an  immense  forest  of  such  oaks,  which,  however,  liter- 
ally grow  into  each  other,  so  that,  by  the  junction  and 
intercresence  of  root  with  root  and  limb  with  limb,  the 
manifold  wood  becomes  a  single  living  organic  whole. 
A  vast  complex  of  interlacing  theories  —  that  the  science 
now  is  actually,  but  it  is  far  more  wondrous  still  poten- 


130  THE   UNIVERSE   AND   BEYOND 

tially,  its  component  theories  continuing  more  and  more 
to  grow  and  multiply  beyond  all  imagination,  and 
beyond  the  power  of  any  single  genius,  however  gifted. 
What  is  this  thing  so  marvellously  vital?  What  does 
it  undertake?  What  is  its  motive?  How  is  it  related 
to  other  modes  and  interests  of  the  human  spirit? 

One  of  the  oldest  and  at  the  same  time  the  most 
familiar  of  the  definitions  conceived  mathematics  to  be 
the  science  of  magnitude,  where  magnitude,  including 
multitude  as  a  special  kind,  was  whatever  was  capable 
of  increase  and  decrease  and  measurement.  This  last  — 
capability  of  measurement  —  was  the  essential  thing. 
That  was  a  most  natural  definition  of  the  science,  for 
magnitude  is  a  singularly  fundamental  notion,  not  only 
inviting  but  demanding  consideration  at  every  stage 
and  turn  of  life.  The  necessity  of  finding  out  how  many 
and  how  much  was  the  mother  of  counting  and  measure- 
ment, and  mathematics,  first  from  necessity  and  then 
from  joy,  so  busied  itself  with  these  things  that  they 
came  to  seem  its  whole  employment.  But  now  the 
notion  of  measurement  as  the  repeated  application  of 
a  constant  unit  has  been  so  refined  and  generalised,  on 
the  one  hand  through  the  creation  of  imaginary  and 
irrational  numbers,  and  on  the  other  by  use  of  a  scale, 
as  in  non-Euclidian  geometry,  where  the  unit  suffers  a 
lawful  change  from  step  to  step  of  its  application,  that 
to  retain  the  old  words  and  call  mathematics  the  science 
of  measurement  seems  quite  inept  as  no  longer  telling 
what  the  spirit  of  mathesis  is  really  bent  upon.  More- 
over, the  most  striking  measurements,  as  of  the  volume 
of  a  planet,  the  swiftness  of  thought,  the  valency  of 
atoms,  the  velocity  of  light,  the  distance  of  star  from 
star,  are  not  achieved  by  direct  repeated  application 
of  a  unit.  They  are  all  accomplished  by  indirection. 


THE   UNIVERSE   AND   BEYOND  13! 

And  it  was  perception  of  this  fact  which  led  to  the 
famous  definition  by  the  philosopher  and  mathematician, 
Auguste  Comte,  that  mathematics  is  the  science  of 
indirect  measurement.  Doubtless  we  have  here  a  finer 
insight  and  a  larger  view,  but  the  thought  is  yet  too 
narrow,  nor  is  it  deep  enough.  For  it  is  obvious  that 
there  is  much  mathematical  activity  which  is  not  at  all 
concerned  with  measurement,  either  direct  or  indirect. 
In  projective  geometry,  for  example,  it  was  observed 
that  metric  considerations  were  by  no  means  chief.  As 
a  simplest  illustration,  the  fact  that  two  points  deter- 
mine a  line,  or  the  fact  that  a  plane  cuts  a  sphere  in  a 
circle,  is  not  a  metric  fact,  being  concerned  with  neither 
size  nor  magnitude.  Here  it  was  position  rather  than 
size  that  seemed  to  some  to  be  the  central  idea,  and 
so  it  was  proposed  to  call  mathematics  the  science  of 
magnitude,  or  measurement,  and  position. 

Even  as  thus  expanded,  the  definition  yet  excludes 
many  a  mathematical  realm  of  vast,  nay,  infinite  extent. 
Consider,  for  example,  that  immense  class  of  things 
familiarly  known  as  operations.  These  are  limitless, 
alike  in  number  and  in  kind.  Now  it  so  happens  that 
there  are  systems  of  operations  such  that  any  two  opera- 
tions of  a  given  system  which  follow  one  another  pro- 
duce the  same  effect  as  some  other  single  operation  of 
the  system.  For  an  illustration,  think  of  all  possible 
straight  motions  in  space.  The  operation  of  going 
from  A  to  B  followed  by  the  operation  of  going  from 
B  to  C  is  equivalent  to  the  single  operation  of  going 
from  A  to  C.  Thus,  the  system  of  such  straight  opera- 
tions is  a  closed  system.  Combination  of  any  two  of 
them  yields  another  operation,  not  without,  but  within 
the  system.  Now  the  theory  of  such  closed  systems  — 
called  groups  of  operations  —  is  a  mathematical  theory, 


132  THE   UNIVERSE   AND   BEYOND 

already  of  colossal  proportions,  and  still  growing  with 
astonishing  rapidity.  But,  and  this  is  the  point,  an 
abstract  operation,  though  a  very  real  thing,  is  neither 
a  position  nor  a  magnitude. 

This  way  of  trying  to  come  at  an  adequate  conception 
of  mathematics,  viz.,  by  naming  its  different  domains, 
or  varieties  of  content,  is  not  likely  to  prove  successful. 
For  it  demands  an  exhaustive  enumeration  not  only  of 
the  fields  now  occupied  by  the  science,  but  also  of  the 
realms  destined  to  be  conquered  by  it  in  the  future,  and 
such  an  achievement  would  require  a  prevision  that  none 
perhaps  could  claim. 

Fortunately  there  are  other  paths  of  approach  that 
seem  more  promising.  Everyone  has  observed  that 
mathematics,  whatever  it  may  be,  possesses  a  certain 
mark,  namely,  a  degree  of  certainty  not  found  else- 
where. So  it  is,  proverbially,  the  exact  science  par 
excellence.  Exact,  you  say,  but  in  what  sense?  To  this 
an  excellent  answer  is  contained  in  a  definition  given 
by  an  American  mathematician,  Professor  Benjamin 
Peirce:  Mathematics  is  the  science  which  draws  necessary 
conclusions,  a  formulation  something  more  than  finely 
paraphrased  by  one  l  of  my  own  teachers  thus :  Mathe- 
matics is  the  universal  art  apodictic.  These  statements, 
though  neither  of  them  may  be  entirely  satisfactory,  are 
both  of  them  telling  approximations.  Observe  that  they 
place  the  emphasis  on  the  quality  of  being  correct.  Noth- 
ing is  said  about  the  conclusions  being  true.  That  is 
another  matter,  to  which  I  will  return  presently.  But 
why  are  the  conclusions  of  mathematics  correct?  Is  it 
that  the  mathematician  has  an  essentially  different 
reasoning  faculty  from  other  folks?  By  no  means. 
What,  then,  is  the  secret?  Reflect  that  conclusion  im- 

1  Professor  W.  B.  Smith. 


THE   UNIVERSE   AND   BEYOND  133 

plies  premises,  and  premises  imply  terms,  and  terms 
stand  for  ideas  or  concepts,  and  that  these,  namely, 
concepts,  are  the  ultimate  material  with  which  the 
spiritual  architect,  which  we  call  the  Reason,  designs 
and  builds.  Here,  then,  we  may  expect  to  find  light. 
The  apodictic  quality  of  mathematical  thought,  the 
certainty  and  correctness  of  its  conclusions,  are  due, 
not  to  a  special  mode  of  ratiocination,  but  to  the  char- 
acter of  the  concepts  with  which  it  deals.  What  is  that 
distinctive  characteristic?  I  answer:  precision,  sharp- 
ness, completeness,  of  definition.  But  how  comes  your 
mathematician  by  such  completeness?  There  is  no  mys- 
terious trick  involved;  some  ideas  admit  of  such  pre- 
cision, others  do  not;  and  the  mathematician  is  one  who 
deals  with  those  that  do.  Law,  says  Blackstone,  is  a 
rule  of  action  prescribed  by  the  supreme  power  of  a 
state  commanding  what  is  right  and  prohibiting  what 
is  wrong.  But  what  are  a  state  and  supreme  power  and 
right  and  wrong?  If  all  such  terms  admitted  of  com- 
plete determination,  then  the  science  of  law  would  be 
a  branch  of  pure,  and  its  practice  a  branch  of  applied, 
mathematics.  But  does  not  the  lawyer  sometimes  arrive 
at  correct  conclusions?  Undoubtedly  he  does  some- 
times, and,  what  may  seem  yet  more  astonishing,  so 
does  your  historian  and  even  your  sociologist,  and  that 
without  the  help  of  accident.  When  this  happens,  how- 
ever, when  these  students  arrive,  I  do  not  say  at  truth, 
for  that  may  be  by  lucky  accident  or  happy  chance  or  a 
kind  of  intuition,  but  when  they  arrive  at  conclusions 
that  are  correct,  then  that  is  because  they  have  been 
for  the  moment  in  all  literalness  acting  the  part  of 
mathematician.  I  do  not  say  that  for  the  aggrandise- 
ment of  mathematics.  Rather  is  it  for  credit  to  all 
thinkers  that  none  can  show  you  any  considerable  gar- 


134  THE   UNIVERSE   AND   BEYOND 

ment  of  thought  in  which  you  may  not  find  here  and 
there,  rarely  enough  sometimes,  a  golden  fibre  woven  in 
some,  it  may  be,  exceptional  moment,  of  precise  con- 
ception and  rigorous  reasoning.  To  think  right  —  that 
is  no  characteristic  striving  of  a  class  of  men.  It  is  a 
common  aspiration.  Only,  the  stuff  of  thought  is  mostly 
intractable,  formless,  like  some  milky  way  waiting  to 
be  analysed  into  distinct  star-forms  of  definite  ideas. 
All  thought  aspires  towards  the  character  and  condition 
of  mathematics. 

The  reality  of  this  aspiration  and  the  distinction  it 
implies  admit  of  many  illustrations,  of  which  here  a 
single  one  must  suffice.  There  is  no  more  common  or 
more  important  notion  than  that  of  function,  the  term 
being  applied  to  either  of  two  variable  things  such  that 
to  any  value  or  state  of  either  there  correspond  one  or 
more  values  or  states  of  the  other.  Of  such  function 
pairs,  examples  abound  on  every  hand,  as  the  radius 
and  the  area  of  a  circle,  the  space  traversed  and  the 
rate  of  going,  progress  of  knowledge  and  enthusiasm  of 
study,  elasticity  of  medium  and  velocity  of  sound  or 
other  undulation,  -the  amount  of  hydrogen  chloride 
formed  and  the  time  occupied,  the  prosperity  of  a 
given  community  and  the  intelligence  of  its  patriotism. 
Indeed,  it  may  very  well  be  that  there  is  nothing  which 
is  not  in  some  sense  a  function  of  every  other.  Be  that 
as  it  may,  one  thing  is  very  certain,  namely,  a  very 
great  part  and  probably  all  of  our  thinking  is  concerned 
with  functional  relationships,  deals,  that  is,  with  pairs 
of  systems  of  corresponding  values  or  states  or  changes. 
Behold,  for  example,  how  the  parallelistic  psychology 
searches  for  correlations  between  psychical  and  physical 
phenomena.  Witness,  too,  the  sociologist  trying  to  de- 
termine the  correspondence  between  the  peacefulness 


THE   UNIVERSE   AND   BEYOND  135 

and  the  homogeneity  of  a  population,  or,  again,  be- 
tween manifestations  of  piety  or  the  spread  of  populism 
and  the  condition  of  the  crops.  It  is  then  here,  in  the 
wondrous  domain  of  correspondence,  the  answering  of 
value  to  value,  of  change  to  change,  of  condition  to 
condition,  of  state  to  state,  that  the  knowing  activity 
finds  its  field. 

What  is  it  precisely  that  we  seek  in  a  correlation? 
The  answer  is:  when  one  or  more  facts  are  given,  to  pass, 
with  absolute  certainty,  to  the  correlative  fact  or  facts.  To 
do  this  obviously  requires  formulae  or  equations  which 
precisely  define  the  manner  of  correlation,  or  the  law 
of  interdependence.  Where  do  such  formulae  come  from? 
I  answer  that,  strictly  speaking,  they  are  never  found, 
they  are  always  assumed.  Now,  nothing  is  easier  than 
to  write  down  a  perfectly  definite  formula  that  does  not 
tell,  for  example,  how  cheerfulness  depends  on  climate, 
or  how  pressure  affects  the  volume  of  a  gas.  Nay,  a 
given  formula  may  be  perfectly  intelligible  in  itself,  it 
may  state,  that  is,  a  perfectly  intelligible  law  of  cor- 
respondence, which,  nevertheless,  may  have  no  validity 
at  all  in  the  physical  universe  and  none  elsewhere  than 
in  the  formula  itself.  What,  then,  guides  in  the  choice 
of  formulae?  That  depends  upon  your  kind  of  curiosity, 
and  curiosity  is  not  a  matter  of  choice. 

Just  here  we  are  in  a  position  where  we  have  only  to 
look  steadily  a  little  in  order  to  see  the  sharp  distinc- 
tion between  mathematics  and  natural  science.  These 
are  discriminated  according  to  the  kind  of  curiosity 
whence  they  spring.  The  mathematician  is  curious  about 
definite  abstract  correspondences,  about  perfectly-defined 
functional  relationships  in  themselves.  These  are  more 
numerous  than  the  sands  of  the  seashore,  they  are  as 
multitudinous  as  the  points  of  space.  It  is  this  as- 


136  THE   UNIVERSE   AND   BEYOND 

semblage  of  pure,  precisely-defined  relationships  which 
constitute  the  mathematician's  universe,  an  indefinitely 
infinite  universe,  worlds  of  worlds  of  wonders,  incon- 
ceivably richer  than  the  outer  world  of  sense.  This 
latter  is  indeed  immense  and  marvellous,  with  its  rolling 
seas  and  stellar  fields  and  undulating  ether,  but,  com- 
pared with  the  hyperspaces  explored  by  the  genius  of  the 
geometrician,  the  whole  vast  extent  of  the  sensuous  uni- 
verse is  a  merest  point  of  light  in  a  blazing  sky. 

Now  this  mere  speck  of  a  physical  universe,  in  which 
the  chemist  and  the  physicist,  the  biologist  and  the  so- 
ciologist, and  the  rest  of  nature  devotees,  find  their 
great  fields,  may  be,  as  it  seems  to  be,  an  organic  thing, 
connected  into  an  ordered  whole  by  a  tissue  of  definable 
functional  relationships,  and  it  may  not.  The  nature 
devotee  assumes  that  it  is  and  tries  to  find  the  relation- 
ships. The  mathematician  does  not  make  that  assump- 
tion and  does  not  seek  for  relationships  in  the  outer 
world.  Is  the  assumption  correct?  As  man,  the  mathe- 
matician does  not  know,  although  he  greatly  cares.  As 
mathematician,  man  neither  knows  nor  cares.  The 
mathematician  does  know,  however,  that,  if  the  assump- 
tion be  correct,  every  definite  relationship  that  is  valid 
in  nature,  every  type  of  order  and  mode  of  correlation 
obtaining  there,  is,  in  itself,  a  thing  for  his  thought,  an 
essential  element  in  his  domain  of  study.  He  knows,  too, 
that,  if  the  assumption  be  not  correct,  his  domain  re- 
mains the  same  absolutely.  The  two  realms,  of  mathe- 
matics, of  nature  science,  are  fundamentally  distinct 
and  disparate  forever.  To  think  the  thinkable  —  that  is 
the  mathematician's  aim.  To  assume  that  nature  is 
thinkable,  an  incarnate  rational  logos,  and  to  seek  the 
thought  supposed  incarnate  there  —  these  are  at  once 
the  principle  and  the  hope  of  the  nature  student.  Sci- 


THE   UNIVERSE   AND   BEYOND  137 

ence,  said  Riemann,1  is  the  attempt  to  comprehend  nature 
by  means  of  concepts.  Suppose  the  nature  student  is 
right,  suppose  the  physical  universe  really  is  an  enfleshed 
logos  of  reason,  does  that  imply  that  all  the  thinkable 
is  thus  incorporated?  It  does  not.  A  single  ordered 
universe,  one  that  through  and  through  is  self -com- 
patible, cannot  be  the  whole  of  reason  materialised  and 
objectified.  There  is  many  a  rational  logos,  and  the 
mathematician  has  high  delight  in  the  contemplation 
of  inconsistent  systems  of  consistent  relationships.  There 
are,  for  example,  a  Euclidean  geometry  and  more  than 
one  species  of  non-Euclidean.  As  theories  of  a  given 
space,  these  are  not  compatible.  If  our  universe  be, 
as  Plato  thought,  and  nature  science  takes  for  granted, 
a  space-conditioned,  geometrised  affair,  one  of  these 
geometries  may  be,  none  of  them  may  be,  not  all  of 
them  can  be,  valid  in  it.  But  in  the  vaster  world  of 
thought,  all  of  them  are  valid,  there  they  co-exist,  and 
interlace  among  themselves  and  others,  as  differing  com- 
ponent strains  of  a  higher,  strictly  supernatural,  hyper- 
cosmic,  harmony. 

It  is,  then,  in  the  inner  world  of  pure  thought,  where 
all  entia  dwell,  where  is  every  type  of  order  and  manner 
of  correlation  and  variety  of  relationship,  it  is  in  this 
infinite  ensemble  of  eternal  verities  whence,  if  there  be 
one  cosmos  or  many  of  them,  each  derives  its  character 
and  mode  of  being,  —  it  is  there  that  the  spirit  of 
mathesis  has  its  home  and  its  life. 

Is  it  a  restricted  home,  a  narrow  life,  static  and  cold 
and  grey  with  logic,  without  artistic  interest,  devoid  of 
emotion  and  mood  and  sentiment?  That  world,  it  is  true, 
is  not  a  world  of  solar  light,  not  clad  in  the  colours  that 

1  C/.  Riemann:  "Fragmcnte  Philosophischcn  Inhalts,"  in  GtsammeUt 
Werke.  These  fragments,  which  are  published  in  English  by  the  Open 
Court  Pub.  Co.,  Chicago,  are  exceedingly  suggestive. 


138  THE   UNIVERSE   AND   BEYOND 

liven  and  glorify  the  things  of  sense,  but  it  is  an  illu- 
minated world,  and  over  it  all  and  everywhere  through- 
out are  hues  and  tints  transcending  sense,  painted  there 
by  radiant  pencils  of  psychic  light,  the  light  in  which  it 
lies.  It  is  a  silent  world,  and,  nevertheless,  in  respect 
to  the  highest  principle  of  art  —  the  interpenetration  of 
content  and  form,  the  perfect  fusion  of  mode  and  mean- 
ing —  it  even  surpasses  music.  In  a  sense,  it  is  a  static 
world,  but  so,  too,  are  the  worlds  of  the  sculptor  and 
the  architect.  The  figures,  however,  which  reason  con- 
structs and  the  mathematical  vision  beholds,  transcend 
the  temple  and  the  statue,  alike  in  simplicity  and  in 
intricacy,  in  delicacy  and  in  grace,  in  symmetry  and  in 
poise.  Not  only  are  this  home  and  this  life  thus  rich 
in  aesthetic  interests,  really  controlled  and  sustained  by 
motives  of  a  sublimed  and  supersensuous  art,  but  the 
religious  aspiration,  too,  finds  there,  especially  in  the 
beautiful  doctrine  of  invariants,  the  most  perfect  sym- 
bols of  what  it  seeks  —  the  changeless  in  the  midst  of 
change,  abiding  things  in  a  world  of  flux,  configurations 
that  remain  the  same  despite  the  swirl  and  stress  of 
countless  hosts  of  curious  transformations.  The  domain 
of  mathematics  is  the  sole  domain  of  certainty.  There 
and  there  alone  prevail  the  standards  by  which  every 
hypothesis  respecting  the  external  universe  and  all  ob- 
servation and  all  experiment  must  be  finally  judged.  It 
is  the  realm  to  which  all  speculation  and  all  thought  must 
repair  for  chastening  and  sanatation  —  the  court  of 
last  resort,  I  say  it  reverently,  for  all  intellection  what- 
soever, whether  of  demon  or  man  or  deity.  It  is  there 
that  mind  as  mind  attains  its  highest  estate,  and  the 
condition  of  knowledge  there  is  the  ultimate  object,  the 
tantalising  goal  of  the  aspiration,  the  Anders-Streben, 
of  all  other  knowledge  of  every  kind. 


THE  AXIOM   OF   INFINITY:    A  NEW 
PRESUPPOSITION  OF  THOUGHT.1 

IT  so  happened  that  when  the  first  number  of  The 
Hibbcrt  Journal  appeared,  containing  an  article  by  Pro- 
fessor Royce  on  the  Concept  of  the  Infinite,  I  had  been 
myself  for  some  time  meditating  on  the  logical  bearings 
and  philosophical  import  of  that  concept,  and  was 
actually  then  engaged  in  marking  out  the  course  which 
it  seemed  to  me  a  first  discussion  of  the  matter  might 
best  follow.  The  order  and  scope  of  his  treatment  were 
so  like  those  I  had  myself  decided  upon  that  I  should 
naturally  have  felt  a  pardonable  pride  in  the  coinci- 
dence, had  not  this  feeling  been  at  the  same  time  quite 
lost  in  a  stronger  one,  namely,  that  of  the  evident 
superiority  of  his  manner  to  any  which  I  could  have 
hoped  to  attain.  Indeed,  so  patient  is  his  exposition 
of  elements,  so  rich  is  it  in  suggestiveness,  so  intimately 
and  instructively,  according  to  his  wont,  has  he  con- 
nected the  most  abstruse  and  recondite  of  doctrines 
with  the  most  obvious  and  seemingly  trivial  of  things, 
and  so  luminous  and  stimulating  is  it  all,  that  one  must 
admire  the  ingenuity  it  betrays,  and  cannot  but  wonder 
whether  after  all  there  really  are  in  science  or  philosophy 
any  notions  too  remote  and  obscure  to  be  rendered 
intelligible  even  to  common  sense,  if  only  a  sufficiently 
cunning  pen  be  engaged  in  the  service. 

1  Appeared  in  The  Hibbert  Journal,  April,  1904. 


140  THE   AXIOM   OF   INFINITY 

While  his  paper  is  thus  replete  with  inspiring  intima- 
tions of  the  "glorious  depths"  and  near-lying  interests 
of  the  doctrine  treated,  and  is,  in  point  of  clearness 
and  vivid  portrayal  of  its  central  thought,  a  model  be- 
yond the  art  of  most,  it  is  not,  I  believe,  equally  happy 
when  judged  on  the  severer  ground  of  its  critico-logical 
estimates.  Even  on  this  ground,  I  do  not  hesitate,  after 
close  examination,  to  adjudge  it  the  merit  of  general 
soundness.  That,  however,  it  is  thoroughly  sound,  com- 
pletely mailed  against  every  possible  assault  of  criti- 
cism, is  a  proposition  I  am  by  no  means  prepared  to 
maintain.  Quite  the  contrary,  in  fact.  Nor  can  the 
defects  be  counted  as  trivial.  One  of  them  especially, 
which  it  has  in  common  with  other  both  earlier  and 
later  discussions  of  the  subject,  notably  that  by  Dede- 
kind  himself  and,  more  recently,  that  by  Mr.  Bertrand 
Russell  in  his  imposing  treatise  on  The  Principles  of 
Mathematics,  is  of  the  most  radical  nature,  concerning 
as  it  does  no  less  a  question  than,  I  do  not  say  merely 
that  of  the  validity,  but  that  of  the  possibility,  of 
existence-proofs  of  the  infinite. 

And  here  I  may  as  well  state  at  once,  lest  there  should 
be  some  misapprehension  in  respect  to  purpose,  that 
the  present  writing  is  not  primarily  designed  to  be  a 
review  of  Professor  Royce's  or  of  other  recent  discus- 
sions of  the  infinite.  Reviewed  to  some  extent  they 
will  be,  but  only  incidentally,  and  mainly  because  they 
have  declared  themselves,  erroneously  as  I  think,  upon 
that  most  fundamental  of  questions,  namely,  whether  it 
is  possible,  by  aid  of  the  modern  concept,  to  demonstrate 
the  existence,  of  the  infinite.  Argument  would  seem 
superfluous  to  show  the  immeasurable  import  of  this 
problem,  whether  it  be  viewed  solely  in  its  immediate 
logical  bearings,  or  also  mediately,  through  the  latter, 


THE   AXIOM   OF   INFINITY  14! 

in  its  bearings  upon  philosophy,  upon  theology,  and, 
only  more  remotely,  upon  religion  itself.  It  is  chief 
among  the  aims  of  this  essay,  to  open  that  problem 
anew,  to  appeal  from  the  prevailing  doctrine  concerning 
it,  in  the  hope  of  securing,  if  possible,  a  readjudication 
of  the  matter  which  shall  be  final. 

This  subject  of  the  infinite,  how  it  baffles  approach! 
How  immediate  and  how  remote  it  seems,  how  it  abides 
and  yet  eludes  the  grasp,  how  familiar  it  appears, 
mingling  with  the  elemental  simplicities  of  the  heart, 
continuously  weaving  itself  into  the  intimate  texture 
of  common  life,  and  yet  how  austere  and  immense  and 
majestic,  outreaching  the  sublimest  flights  of  the  im- 
agination, transcending  the  stellar  depths,  immeasurable 
by  the  beginningless,  endless  chain  of  the  ages!  Com- 
prehend the  infinite!  No  wonder  we  hear  that  none 
but  the  infinite  itself  is  adequate  to  that.  Du  gleichst 
dem  Gtist,  den  du  begreifsl.  Be  it  so.  Perhaps,  then, 
we  are  infinite.  If  not, 

"'Wie'  fass'  ich  dich,  unendliche  Natur?" 

Or  is  it  finally  a  mere  illusion?  And  is  there  after  all  no 
infinite  reality  to  be  seized  upon?  Again,  if  not,  what 
signifies  the  finite?  Is  that  to  be  for  ever  without 
definition,  except  as  reciprocal  of  that  which  fails  to 
be?  Is  the  All  really  enclosed  in  some  vast  ellipsoid, 
without  a  beyond,  incircumscriptible,  devoid  alike  of 
tangent  plane  and  outer  point?  Are  we  eternally  con- 
demned to  seek  therein  for  the  meaning  and  end  of 
processes  that  refuse  to  terminate?  And  is,  then,  this 
region,  too,  but  a  locus  of  deceptions,  "of  false  alluring 
jugglery"?  Is  analysis  but  the  victim  of  hallucination 
when  it  thinks  to  detect  the  existence  of  realms  that 
underlie  and  overarch  and  compass  about  the  domain  of 


142  THE   AXIOM  OF   INFINITY 

the  countable  and  measurable?  And  does  the  spirit, 
in  its  deeper  musings,  in  its  pensive  moods,  only  seem  to 
feel  the  tremulous  touch  of  transfinite  waves,  of  vitalising 
undulations  from  beyond  the  farthest  shore  of  the  sea 
of  sense? 

One  fact  at  once  is  clear,  namely,  that,  whatever  ulti- 
mate justification  the  hypothesis  may  find,  thought  has 
never  escaped  the  necessity  of  supposing  the  universe 
of  things  to  be  intrinsically  somehow  cleft  asunder  into 
the  two  Grand  Divisions,  or  figured,  if  you  will,  under 
the  two  fundamental  complementary  all-inclusive  Forms, 
which,  from  motives  more  or  less  distinctly  felt  and  also 
just,  as  we  shall  see,  though  not  quite  justified,  have 
been,  from  time  immemorial,  designated  as  the  Finite 
and  the  Infinite.  And  these  great  terms  or  their  verbal 
equivalents  —  for  concepts  in  any  strict  sense  they  have 
not  been  —  though  always  vague  and  shifting,  for  ever 
promising  but  never  quite  delivering  the  key  to  their 
identities  into  the  hand  of  Definition,  have,  neverthe- 
less, in  every  principal  scene,  together  played  the  gravest 
role  in  the  still  unfolding  drama  of  speculation.  Or,  to 
change  the  figure,  they"  have  been  as  Foci,  one  of  them 
seemingly  near,  the  other  apparently  remote,  neither 
of  them  quite  itself  determinate,  but  the  two  con- 
jointly serving  always  to  determine  the  ever-varying 
eccentricity  of  the  orbit  of  thought;  and  doubtless  the 
vaster  lines  that  serve  to  bind  the  differing  epochs  of 
speculation  into  a  single  continuous  system  can  best  be 
traced  by  reference  to  these  august  terms  as  co-ordinate 
poles  of  interest. 

As  a  simple  historical  fact,  then,  philosophy  has  indeed, 
with  but  negligible  exception,  throughout  assumed  the 
existence  of  both  the  finite  and  the  infinite.  That  is  one 
thing.  Another  fact  of  distinct  and  equal  weight,  no 


THE   AXIOM  OF   INFINITY  143 

matter  whether  or  how  we  may  account  for  it,  is  that 
man,  in  accord  with  the  deeper  meaning  of  the  Pro- 
tagorean  maxim,  has  always  felt  himself  to  have  within, 
or  to  be  somehow,  the  potential  measure  of  all  that  is. 
Is  it  insignificant  that  this  faith  —  for  that  is  what  it 
seems  to  be  —  as  if  an  indestructible  character  of  the 
race,  as  if  an  invariant  defining  property  of  the  germ 
plasm  itself  whence  man  springs  and  derives  his  con- 
tinuity, should  have  survived  every  vicissitude  of  human 
fortune?  that  it  should  have  been  indeed,  if  not  the 
substance,  at  least  the  promise,  of  things  hoped  for,  the 
evidence,  too,  of  things  not  seen,  marking  and  sustain- 
ing metaphysical  research  from  the  earliest  times?  And, 
what  is  more,  the  spirit  of  such  research,  curiosity  I 
mean,  fit  companion  and  counterpart  of  that  abiding 
faith,  unlike  "experience  and  observation,"  has  known 
no  bounds,  but,  on  the  contrary,  finding  within  itself 
no  fatal  principle  of  limitation,  it  has  ever  disdained  the 
scale  of  finite  things  as  competent  to  take  its  measure, 
and  boldly  asserted  claim  to  the  entire  realm  of  being. 

These  questions,  however,  have  been  something  more 
than  fascinating.  Perhaps  their  rise,  but  not  their  mani- 
fold development,  much  less  their  profound  significance 
for  life  and  thought,  is  to  be  adequately  explained  on 
the  hypothesis  of  insatiate  curiosity  alone.  It  must  be 
granted  that  their  presence,  especially  in  the  arena  of 
dialectic,  has  been  often  due  simply  to  their  intrinsic 
magical  charm  for  "summit-intellects."  And  doubtless 
the  play-instinct,  deep-dwelling  in  the  constitution  of 
the  mind,  has  often  made  them  serve  the  higher  faculties 
merely  as  intricate  puzzles,  to  beguile  the  time  withal. 
But,  in  general,  the  questions  have  worn  a  sterner 
aspect.  Philosophy  has  been  not  merely  allured,  it  has 
been  constrained,  to  their  consideration;  constrained  not 


144  THE   AXIOM   OF   INFINITY 

only  because  of  their  inherence  in  problems  of  the  con- 
science, especially  in  that  most  radical  problem  of  find- 
ing the  simplest  system  of  postulates  that  shall  be  at 
once  both  necessary  and  sufficient  to  explain  the  moral 
feeling;  but  constrained  still  more  powerfully  by  the 
insistent  demands  that  issue  from  the  religious  con- 
sciousness. But  this  is  yet  not  all.  For  man  cannot 
live  by  these  august  interests  alone.  And  it  is  pro- 
foundly significant,  both  as  witnessing  to  the  final  inter- 
blending,  the  fundamental  unity,  of  all  the  concerns  of 
the  human  spirit,  and  as  revealing  the  ultimate  depth 
and  dignity  of  all  its  interests,  that  questions  about  the 
infinite  quite  similar  to  those  that  claim  so  illustrious 
parentage  in  Ethics  and  Philosophy,  admit  elsewhere  of 
humbler  derivation,  and  readily  own  to  the  lowliest  of 
origins.  Man,  indeed,  merely  to  live,  has  had  to  meas- 
ure and  to  count,  and  this  homely  necessity,  fruitful 
mother  of  mystery  and  doubt,  independently  set  the 
problems  of  the  indefinitely  small  and  the  indefinitely 
great;  and  so  it  was  that  needs  quite  as  immediate  and 
austere  as  those  of  Morals  and  Religion  —  I  mean  the 
exigencies  of  Science,  arid  especially  of  Mathematics  — 
demanded  on  their  own  ground,  in  the  very  beginnings 
of  exact  knowledge,  that  the  understanding  transcend 
every  possible  sequence  of  observations,  pass  the  utter- 
most limit  of  "experience,"  which,  refine  and  enlarge 
it  as  you  may,  remains  but  finite,  and  literally  lay  hold 
on  infinity  itself. 

To  this  ancient  irrevocable  demand,  thus  urged  upon 
the  reason  from  every  cardinal  point  of  human  interest, 
genius  has  responded  as  to  a  challenge  from  the  gods, 
and  I  submit  that  the  response,  the  endeavour  of  the 
reason  actually  to  subjugate  extra-finite  being  and  com- 
pel surrender  of  its  secrets  by  the  organon  of  thought, 


THE   AXIOM  OP   INFINITY  145 

constitutes  the  most  sublime  and  strenuous  and  inspiring 
enterprise  of  the  human  intellect  in  every  age. 

What  of  it?  Long  centuries  of  gigantic  striving,  age 
on  age  of  philosophic  toil,  immeasurable  devotion  of 
time  and  energy  and  genius  to  a  single  end,  the  intel- 
lectual conquest  of  transfmite  being  —  what  has  it  all 
availed?  What  triumphs  have  been  won?  I  speak, 
narrowly,  of  the  conquest,  and  demand  to  know,  not 
whether  it  has  been  accomplished  —  for  that  were  a 
foolish  query  —  but  whether,  strictly  speaking,  it  has 
been  begun.  Let  not  the  import  of  the  question  be  mis- 
taken. No  answer  is  sought  in  terms  of  such  moral  or 
"spiritual"  gains  as  may  be  incident  even  to  efforts  that 
miss  their  aim.  Everyone  knows  that  seeking  has  com- 
pensations of  its  own,  which  indeed  are  ofttimes  better 
than  any  which  finding  itself  can  give.  And  it  seems 
sometimes  as  if  the  higher  life  were  chiefly  sustained  by 
unsought  gains  incident  to  the  unselfish  pursuit  of  the 
unattainable.  The  circle  has  not  been  squared,  nor  the 
quintic  equation  solved,1  nor  perpetual  motion  invented; 
neither  indeed  can  be;  yet  it  would  show  but  meagre 
understanding  of  the  ways  of  truth  to  men,  did  one 
suppose  all  the  labour  devoted  to  such  problems  to  have 
been  without  reward.  So,  conceivably,  it  might  be 
with  this  problem  of  the  infinite.  It  may  be  granted 
that,  even  supposing  no  solution  to  be  attainable,  the 
ceaseless  search  for  one,  the  unwearied  high  endeavor 
of  the  reason  through  the  ages,  presents  a  spectacle 
ennobling  to  behold,  and  of  which  mankind,  it  may  be, 
could  ill  afford  to  be  deprived.  It  may  be  granted  that 
incidentally  many  insights  have  been  won  which,  though 
not  solutions,  have  nevertheless  permanently  enriched 
the  literature  of  the  world  and  are  destined  to  improve 
1  That  is,  by  means  of  radicals. 


146  THE   AXIOM   OF   INFINITY 

its  life.  It  may  be  granted  that  in  every  time  some 
doctrine  of  infinity,  some  philosophy  of  it,  has  been  at 
least  effective,  has  helped,  that  is,  for  better  or  worse, 
to  fashion  the  forms  of  human  institutions  and  to  de- 
termine the  course  of  history.  Concerning  none  of 
these  things  is  there  here  any  question.  As  to  what  the 
question  precisely  is,  there  need  not  be  the  slightest  mis- 
apprehension. The  fact  is  that  for  thousands  of  years 
philosophy  has  recognised  the  presence  of  a  certain 
definite  Problem,  namely,  that  of  extending  the  dominion 
of  logic,  the  reign  of  exact  thought,  out  beyond  the  utmost 
reach  of  finite  things  into  and  over,  the  realm  of  infinite 
being,  and  this  problem,  by  far  the  greatest  and  most 
impressive  of  her  strictly  intellectual  concernments, 
philosophy  has,  for  thousands  of  years,  arduously  striven 
to  solve.  And  now  I  ask  —  not,  has  it  been  worth  while? 
for  that  is  conceded,  but  —  has  she  advanced  the  solu- 
tion in  any  measure,  and,  if  so,  in  what  respect,  and  to 
what  extent? 

We  are  here  upon  the  grounds  of  the  rational  logos. 
The  whole  force  and  charge  of  the  question  is  directed 
to  matter  of  concept  and  inference.  Fortunately,  the 
answer  is  to  be  as  unmistakable  as  the  question.  It 
must  be  recognised,  of  course,  that  the  "problem,"  as 
stated,  is  exceedingly,  almost  frightfully,  generic,  com- 
prising a  host  of  interdependent  problems.  One  of  these, 
however,  is  pre-eminent:  without  its  solution  none  other 
can  be  solved;  with  its  solution,  any  other  may  be 
eventually.  That  problem  is  the  problem  of  concep- 
tion, of  definition  in  the  unmitigated  rigour  of  its 
severest  meaning;  it  is  the  problem  of  discovering  a 
certain  principle,  of  finding,  without  the  slightest  possi- 
bility of  doubt  or  indetermination,  the  intrinsic  line 
of  cleavage  that  parts  the  universe  of  being  into  its 


THE   AXIOM   OF   INFINITY  147 

two  grandest  divisions,  and  so  of  telling  finally  and 
once  for  all  precisely  what,  for  thought,  the  infinite  is 
and  what,  for  thought,  the  finite  is. 

And  now,  thanks  to  the  subtle  genius  of  the  modern 
Teutonic  mind,  this  ancient  problem,  having  baffled  the 
thought  of  all  the  centuries,  has  been  at  last  com- 
pletely solved,  and  therein  our  original  question  finds 
its  answer:  The  conquest  has  been  begun.  Bernhard 
Riemann,  profound  mathematician  and  —  important 
fact,  of  which,  strangely  enough,  too  many  philosophers 
seem  invincibly  unaware  —  profound  metaphysician  too, 
having  pointed  out,  in  his  famous  Habilitationschrift* 
the  epoch-making  distinction  between  mere  boundless- 
ness and  infinitude  of  manifolds  similar  to  that  of  space, 
the  greater  glory  was  reserved  for  three  contemporary 
compatriots  of  his  —  Bernhard  Bolzano,*  Richard  Dede- 
kind,*  and  Georg  Cantor,4  the  first  an  acute  and  learned 
philosopher  and  theologian,  with  deep  mathematical 
insight,  the  other  two  brilliant  mathematicians,  with  a 
strong  bent  for  metaphysics  —  to  win  independently 
and  about  the  same  time  the  long-coveted  insight  into 
the  intrinsic  nature  of  infinity.  And  thus  it  is  a  dis- 
tinction of  our  own  time  that  within  the  memory  of 
living  men  the  defining  mark  of  the  infinite  first  failed 
to  elude  the  grasp,  and  that  that  august  term,  after 
the  most  marvellous  career  of  any  in  the  history  of 
speculation,  has  been  finally  made  to  assume  the  prosaic 
form  of  an  exact  and  completely  determined  concept, 
and  so  at  length  to  become  available  for  the  purposes  of 
rigorously  logical  discourse. 

1  "Ueber  die  Hypothesen,  wclche  die  Geometric  zu  Grande  liegen," 
Gts.  Werhe.    Also  in  English  by  W.  R.  Clifford. 
*  "Paradoxicn  ties  Unendlichen." 
i  ""Was  sind  und  was  sollen  die  Zahlen." 
4  Memoirs  in  Ada  Mathematics,  vol.  ii.,  and  elsewhere. 


148  THE   AXIOM   OF   INFINITY 

Pray,  then,  what  is  this  concept?  Of  various  equiva- 
lent forms  of  statement,  I  choose  the  following:  An 
assemblage  (ensemble,  collection,  group,  manifold)  of  ele- 
ments (things,  no  matter  what]  is  infinite  or  finite  accord- 
ing as  it  has  or  has  not  a  PART  to  which  the  whole  is  just 
EQUIVALENT  in  the  sense  that  between  the  elements  com- 
posing that  part  and  those  composing  the  whole  there  subsists 
a  unique  and  reciprocal  (one-to-one)  correspondence. 

If  we  may  trust  to  intuition  in  questions  about  reality, 
assemblages,1  infinite  as  defined,  actually  abound  on 
every  hand.  I  need  not  pause  to  indicate  examples. 
Those  pointed  out  in  Professor  Royce's  mentioned  paper 
may  suffice;  they  will,  at  all  events,  furnish  the  reader 
with  the  "clew,  which,  once  familiar  to  his  hand,  will 
lengthen  as  he  goes,  and  never  break."  The  concept 
itself  I  regard  as  a  great  achievement,  one  of  the  very 
greatest  in  the  history  of  thought.  Not  only  does  it 
mark  the  successful  eventuation  of  a  long  and  toilsome 
search;  it  furnishes  criticism  with  a  new  standard  of 
judgment,  it  at  once  creates,  and  gives  the  means  of 
meeting,  the  necessity  for  a  re-examination  and  a  juster 
evaluation  of  historic  doctrines  of  infinity;  and  it  is 
greater  still,  I  believe,  as  a  destined  instrument  of  ex- 
ploration in  that  realm  which  it  has  opened  to  the  under- 
standing and  whose  boundary  it  defines. 

Is  that  judgment  not  extravagant?  For  the  concept 
seems  so  simple,  is  so  apparently  independent  of  difficult 
presuppositions,  that  one  cannot  but  wonder  why  it  was 
not  formed  long  ago.  Had  the  concept  in  question  been 
early  formed,  the  history  and  present  status  of  philoso- 

1  The  very  simplest  possible  example  of  such  a  manifold  is  that  of  the 
count-numbers.  The  whole  collection  can  be  paired  in  one-to-one  fashion 
with,  for  example,  half  the  collection,  thus:  i,  2;  2,  4;  3,  6;  .  .  .  ;  the 
totality  of  even  and  odd  being  just  equivalent  to  the  even. 


THE   AXIOM  OP  INFINITY  149 

phy  and  theology,  and  of  science  too,  had  doubtless 
been  different.  But  it  was  not  then  conceived.  Now 
that  we  have  it,  is  it  too  unbewildering  to  be  impress- 
ive? Shall  we  esteem  it  lightly  just  because  we  can 
comprehend  it,  because  it  does  not  mystify?  Simple 
it  is  indeed,  almost  as  simple  as  the  Newtonian  law  of 
gravitation,  nearly  as  easy  to  understand  as  the  geo- 
metric interpretation  of  imaginary  quantities,  hardly 
more  difficult  to  grasp  than  the  notion  of  the  conserva- 
tion of  energy,  the  Mendelian  principle  of  inheritance, 
or  than  a  score  of  other  central  concepts  of  science. 
But  shallow  indeed  and  foolish  is  that  criticism  which 
values  ideas  according  to  their  complexity,  and  con- 
founds the  simple  with  the  trivial. 

As  an  immense  city  or  a  vast  complex  of  mountain 
masses,  seen  too  near,  is  obscured  as  a  whole  by  the 
prominence  of  its  parts,  so  the  larger  truth  about  any 
great  subject  is  disclosed  only  as  one  beholds  it  at  a 
certain  remove  which  permits  the  assembling  of  principal 
features  in  a  single  view,  and  a  proportionate  mingling 
of  reflected  light  from  its  grander  aspects.  Accordingly 
it  has  seemed  desirable,  in  the  foregoing  preliminary 
survey,  to  hold  somewhat  aloof,  to  conduct  the  move- 
ment, in  the  main,  along  the  path  of  perspective  centres, 
in  order  to  allow  the  vision  at  every  point  the  amplest 
range.  It  is  now  proposed  to  draw  a  little  closer  to  the 
subject  and  to  examine  some  of  its  phases  more  minutely. 
In  respect  to  the  modern  concept  of  infinity,  we  desire 
to  know  more  fully  what  it  really  signifies,  we  wish  to 
be  informed  how  it  orients  itself  among  cardinal  prin- 
ciples and  established  modes  of  thought.  But  recently 
born  to  consciousness,  it  has  already  been  advanced  to 
conspicuous  and  commanding  station  among  funda- 
mental notions,  and  we  are  concerned  to  know  what,  if 


150  THE   AXIOM   OF   INFINITY 

any,  transformations  of  existing  doctrine,  what  read- 
justments of  attitude  towards  the  universe  without  us 
or  within,  what  changes  in  our  thought  on  ultimate 
problems  of  knowledge  and  reality,  it  seems  to  demand 
and  may  be  destined  to  effect.  In  a  word,  and  speak- 
ing broadly,  we  wish  to  know  not  merely  in  a  narrow 
sense  what  the  new  idea  is,  but,  in  the  larger  meaning 
of  the  term,  what  it  "can." 

I  shall  first  speak  briefly  of  the  so-called  "positive" 
character  of  the  definition,  an  alleged  essential  quality 
of  it,  a  seeming  property  which  criticism  is  wont  to 
signalise  as  a  radical  or  intrinsic  virtue  of  the  concept 
itself.  Quite  independently  of  the  mathematicians 
Dedekind  and  Cantor,  who,  we  have  seen,  were  the 
independent  originators  of  the  new  formulation,  the 
then  old  philosopher,  Bolzano,  bringing  to  the  subject 
another  order  of  training  and  of  motive,  arrived  at 
notions  of  the  finite  and  infinite,  which  on  critical 
examination  are  found  to  be  essentially  the  same  as 
theirs,  though  greatly  differing  in  point  alike  of  view 
and  of  form.  Bolzano's  procedure  is  virtually  as  fol- 
lows:—  Suppose  given  a  class  C  of  elements,  or  things, 
of  any  kind  whatsoever,  as  the  sands  of  the  seashore, 
or  the  stars  of  the  firmament,  or  the  points  of  space, 
or  the  instants  in  a  stretch  of  time,  or  the  numbers 
with  which  we  count,  or  the  total  manifold  of  truths 
known  to  an  omniscient  God.  Out  of  any  such  class  C, 
suppose  a  series  formed  by  taking  for  first  term  one  of 
the  elements  of  C,  for  second  term  two  of  them,  and 
so  on.  Any  term  so  obtainable  is  itself  obviously  a 
class  or  group  of  things,  and  is  defined  to  be  finite.  The 
indicated  process  of  series  formation,  if  sufficiently  pro- 
longed, will  either  exhaust  C  or  it  will  not.  If  it  will, 
C  is  itself  demonstrably  finite;  if  it  will  not,  C  is,  on 


THE  AXIOM  OF  INFINITY  151 

that  account,  defined  to  be  infinite.  Now,  say  Professor 
Royce  and  others,  a  definition  like  the  latter,  being 
dependent  on  such  a  notion  as  that  of  inexhaustibility 
or  endlessness  or  boundlessness,  is  negative;  a  certain 
innate  craving  of  the  understanding  remains  unsatisfied, 
we  are  told,  because  the  definition  presents  the  notion, 
not  in  a  positive  way  by  telling  us  what  the  infinite 
actually  is,  but  merely  in  a  negative  fashion  by  telling 
us  what  it  is  not.  Undoubtedly  the  claim  is  plausible, 
but  is  it  more?  Bolzano  affirmed  and  exemplified  a 
certain  proposition,  in  itself  of  the  utmost  importance, 
and  throwing  half  the  needed  light  upon  the  question 
in  hand.  That  proposition  is:  Any  class  or  assemblage 
(of  elements),  if  infinite  according  to  kis  own  definition 
of  the  term,  enjoys  the  property  of  being  equivalent,  in  the 
sense  above  explained,  to  some  proper  part  of  itself.  Though 
he  did  not  himself  demonstrate  the  proposition,  it 
readily  admits  of  demonstration,  and,  since  his  time, 
has  in  fact  been  repeatedly  and  rigorously  proved. 
Not  only  that,  but  the  converse  proposition,  giving  the 
other  half  of  the  needed  light,  has  been  established  too: 
Every  assemblage  that  HAS  a  part  "equivalent"  to  the 
whole,  is  infinite  in  the  Bolzano  sense  of  the  term. 

It  so  appears,  in  the  conjoint  light  of  those  two 
theorems,  that  the  property  seized  upon  and  pointed 
out  by  the  ingenious  theologian  is  in  all  strictness  a 
characteristic,  though  derivative,  mark  of  the  infinite 
as  he  conceived  and  defined  it.  It  is  sufficiently  obvious, 
therefore,  that  this  derivative  property  might  logically 
be  regarded  as  primitive,  made  to  serve,  that  is,  as  a 
ground  of  definition.  Precisely  this  fact  it  is  which  was 
independently  perceived  by  Dedekind  and  Cantor,  with 
the  result  that,  as  they  have  presented  the  matter,  a 
collection,  or  manifold,  is  infinite  if  it  has  a  certain 


152  THE'  AXIOM   OF   INFINITY 

property,  and  finite  if  it  has  it  not.  And  now,  the  critics 
tell  us,  it  is  the  infinite  which  is  positive  and  the  finite 
which  is  negative. 

The  distinction  appears  to  me  to  be  entirely  devoid 
of  essential  merit.  It  seems  rather  to  be  only  another 
interesting  example  of  that  verbal  legerdemain  for  which 
a  certain  familiar  sort  of  philosophising  has  long  been 
famous.  For  what  indeed  is  positive  and  what  negative? 
Are  we  to  understand  that  these  terms  have  absolute 
as  distinguished  from  relative  meaning?  The  distinc- 
tion, I  take  it,  is  without  external  validity,  is  entirely 
subjective,  a  matter  quite  at  will,  being  dependent 
solely  on  an  arbitrary  ordering  of  our  thought.  That 
which  is  first  put  in  thought  is  positive:  the  opposite, 
being  subsequently  put,  is  negative;  but  the  sens  of  the 
time-vector  joining  the  two  may  be  reversed  at  the 
thinker's  will.  It  is  sometimes  contended  that  that 
which  generally  happens  in  the  world,  and  so  constitutes 
the  rule,  is  intrinsically  positive.  As  a  matter  of  fact 
a  moving  body  "in  general"  continuously  changes  its 
distance  from  every  object.  Such  change  of  distance 
from  every  other  object  would  accordingly  be  a  positive 
something.  Then  it  would  follow  that  the  classic  defi- 
nition of  a  sphere-surface  as  the  locus  of  a  moving 
point  which  does  not  change  its  distance  from  a  certain 
specified  point,  is  really  negative.  Obviously  it  avails 
nothing  essential  to  disguise  the  negativity  by  some 
such  seemingly  positive  phrase  as  "constant"  distance. 
The  trick  is  an  easy  one.  If,  again,  it  be  allowed  that, 
a  process  being  once  started,  its  continuation  is  positive, 
its  termination  negative,  then  it  would  result  that  in- 
exhaustibility is  positive  and  exhaustibility  negative, 
whence  we  should  have  to  own  that  it  is  Bolzano's 
definition  which  is  positive  and  that  by  Dedekind  and 


THE  AXIOM  OP   INFINITY  153 

Cantor  negative.  It  hardly  admits  of  doubt  that  the 
matter  is  purely  one  of  an  arbitrarily  chosen  point  of 
view.  The  distinction  is  here  of  no  importance.  What 
is  important  is  that,  no  matter  which  of  the  definitions 
be  adopted  as  such,  the  other  then  states  a  derivable  prop- 
erty of  the  thing  defined.  In  either  case  the  concept  of 
the  infinite  remains  the  same,  it  is  merely  its  garb  that  is 
changed.  I  am  very  far  from  intending,  however,  to  assert 
herewith  that,  because  the  definitions  are  logically  equiva- 
lent, they  must  needs  be  or  indeed  are  so  practically,  that 
is,  as  instruments  of  investigation.  That  is  another 
matter,  which,  I  regret  to  say,  our  somewhat  pretentious 
critiques  of  scientific  method  furnish  no  better  means  of 
settling  than  the  wasteful  way  of  trial.  Everyone  will 
recall  from  his  school-days  Euclid's  definition  of  a  plane 
as  being  a  surface  such  that  a  line  joining  any  two 
points  of  the  surface  lies  wholly  in  the  surface.  Logi- 
cally that  is  equivalent  to  saying:  A  plane  is  such  an 
assemblage  of  points  that,  any  three  independent  points 
of  the  assemblage  being  given,  one  and  only  one  third 
point  of  the  assemblage  can  be  found  which  is  equi- 
distant from  the  given  three.  But,  despite  their  logical 
equivalence,  who  would  contend  that,  for  elementary 
purposes,  the  latter  notion  is  "practically"  as  good  as 
the  Greek?  And  so  in  respect  to  the  infinite,  I  am  free 
to  admit,  or  rather  I  affirm,  that,  on  the  score  of 
usability,  the  Dedekind-Cantor  definition  is  greatly 
superior  to  its  Bolzanoan  equivalent.  Professor  Roycc 
has  indeed  ingeniously  shown  how  readily  it  lends  itself 
to  philosophic  and  even  to  theologic  uses. 

I  turn  now  to  the  current  assertion  by  Professor 
Royce  and  Mr.  Russell,  that  the  modern  concept  of  the 
infinite,  of  which  I  have  given  above  in  italics  an  exact 
statement,  to  which  the  reader  is  referred,  in  fact  denies 


154  THE   AXIOM   OF   INFINITY 

a  certain  ancient  axiom  of  common  sense,  namely,  the 
axiom  of  whole  and  part.  I  am  not  about  to  submit 
a  brief  in  behalf  of  the  traditional  conception  of  axioms 
as  self-evident  truths.  That  conception,  as  is  well 
known,  has  been  once  for  all  abandoned  by  philosophy 
and  science  alike,  while  to  mathematicians  in  particular 
no  phenomenon  is  more  familiar  than  that  of  the  co- 
existence of  self-coherent  bodies  of  doctrine  constructed 
on  distinct  and  self-consistent  but  incompatible  systems 
of  postulates.  The  co-ordination  of  such  incompatible 
theories  is  quite  legitimate  and  presents  no  cause  for 
regret  or  alarm.  The  forced  recession  of  the  axioms 
from  the  high  ground  of  absolute  authority,  so  far  from 
indicating  chaos  of  intellection  or  ultimate  dissolution 
of  knowledge,  signifies  a  corresponding  deepening  of 
foundation;  it  means  an  ascension  of  mind,  the  procla- 
mation of  its  creative  power,  the  assertion  of  its  own 
supremacy.  And  henceforth  the  denial  of  specific 
axioms,  or  the  deliberate  substitution  of  one  set  for 
another,  is  to  be  rightly  regarded  as  an  inalienable 
prerogative  of  a  liberated  spirit.  The  question  before 
us,  then,  is  one  merely  x>f  fact,  namely,  whether  a  certain 
axiom  is  indeed  denied  or  contradicted  by  the  modern 
concept  of  the  infinite. 

It  is  in  the  first  place  to  be  observed  that  the  statement 
itself  of  that  concept  avoids  the  expression,  "  equality 
of  whole  and  part,"  but  instead  of  it  deliberately  employs 
the  term  "equivalence."  The  word  actually  used  by 
Dedekind  himself  is  dhnlichkeit  (similarity).  But,  says 
Professor  Royce,  "equivalence"  is  just  what  the  axiom 
really  means  by  equality.  It  is  precisely  this  statement 
which  I  venture  to  draw  in  question.  If  we  know  that 
each  soldier  of  a  company  marching  along  the  street 
has  one  and  but  one  gun  on  his  shoulder,  then,  we  are 


THE  AXIOM  OF   INFINITY  155 

told,  even  if  we  do  not  know  how  many  soldiers  or  guns 
there  are,  we  do  know  that  there  are  "as  many"  soldiers 
as  guns.  What  the  definition  in  question,  taken  severely, 
itself  affirms  in  this  case,  is  that  the  assemblage  of  guns 
is  "equivalent  or  similar"  to  that  of  the  soldiers.  Let 
us  now  suppose  that  in  place  of  soldiers  we  write,  for 
example,  "all  positive  integers,"  and  in  place  of  guns, 
"all  even  positive  integers"  -the  integers  are  plainly 
susceptible  of  unique  and  reciprocal  association  with 
the  even  integers,  —  then  the  definition  again  asserts,  as 
before,  "equivalence"  of  these  assemblages.  Note  that 
thus  far  nothing  has  been  said  about  number  as  an 
expression  of  how  many.  If  there  be  a  number  that  tells 
how  many  things  there  are  in  one  assemblage,  that  same 
number  doubtless  tells  how  many  there  are  in  any 
"equivalent"  assemblage,  and  just  because  the  number, 
if  there  be  one,  is  the  same  for  both,  the  two  are  said 
to  be  equal  by  axiom.  In  this  view,  equality  of  groups 
means  more  than  mere  "equivalence";  it  means,  besides, 
sameness  of  their  numbers,  and  so  applies  only  in  case 
there  be  numbers.  But  common  sense,  whose  axiom  is 
here  in  court,  has  neither  found,  nor  affirmed  the  ex- 
istence of,  a  number  telling,  for  example,  how  many 
integers  there  are.  On  the  other  hand,  in  case  of 
assemblages  for  which  common  sense  has  known  a 
number,  the  axiom  of  whole  and  part  is  admittedly 
valid  without  exception.  It  thus  appears  that  the 
axiom  supposed,  regarded,  however  unconsciously  but 
nevertheless  in  intention,  as  applicable  only  in  case 
there  be  a  number  telling  how  many,  is,  in  all  strictness, 
not  denied  by  the  concept  in  question.  Numbers 
designed  to  tell  how  many  elements  there  are  in  an 
assemblage  having  a  part  "equivalent"  to  the  whole 
are  of  recent  invention,  and  it  may  be  remarked  in 


156  THE   AXIOM   OF   INFINITY 

passing  that  this  invention  bears  immediate  favourable 
witness  to  the  fruitfulness  of  the  new  idea.  Such  trans- 
finite  numbers  once  created,  then  undoubtedly,  and 
not  before,  the  question  naturally  presents  itself  whether 
"equivalence"  shall  be  translated  "equality,"  or,  what 
is  tantamount,  whether  the  latter  term  shall  be  gen- 
eralised into  the  former;  "generalised,"  I  say,  for, 
though  it  is  true  that,  as  soon  as  the  transfinite  numbers 
are  created,  there  is,  in  case  of  an  infinite  collection 
and  some  of  its  parts,  a  conjunction  of  "equivalence" 
and  "sameness  of  number,"  yet  equality  does  not  of 
itself  deductively  attach,  for  the  transfinite  numbers 
are  in  genetic  principle,1  i.e.,  radically,  different  from  the 
number  notion  which  the  concept  of  equality  has  hith- 
erto connoted.  The  question  as  to  the  mentioned  trans- 
lation or  generalisation  is,  therefore,  a  question,  and  it 
is  to  be  decided,  not  under  spur  or  stress  of  logic,  but 
solely  from  motives  of  economy  acting  on  grounds  of 
pure  expedience.  If  the  decision  be,  as  seems  likely 
because  of  its  expedience  and  economy  favourable  to 
such  translation  or  generalisation,  then  indeed  the  old 
axiom,  as  above  construed,  still  remains  uncontradicted, 
is  yet  valid  within  the  domain  of  its  asserted  validity. 
It  is  merely  that  a  new  number-domain  has  been  ad- 
joined which  the  old  verity  never  contemplated,  and 
in  which,  therefore,  though  it  does  not  apply,  it  never 
essentially  pretended  to;  but  on  account  of  which  ad- 
junction, nevertheless,  for  the  sake  of  good  neighbour- 
ship, it  is  constrained,  not  indeed  to  retract  its  ancient 
claims,  but  merely  to  assert  them  more  cautiously  and 
diplomatically,  in  preciser  terms.  Even  then,  in  case 
of  quarrel,  it  is  the  generaliser  who  should  explain,  and 
not  a  defender  of  the  generalised. 

1  Cf.  Couturat,  L'Infini  mathtmatiqiw.  Appendix. 


THE  AXIOM   OF   INFINITY  157 

And  now  to  my  final  thesis  I  venture  to  invite  the 
reader's  special  attention,  and  beg  to  be  held  with  utmost 
strictness  accountable  for  my  words.  The  question  is, 
whether  it  is  possible,  by  means  of  the  new  concept,  to 
demonstrate  the  existence  of  the  infinite;  whether,  in 
other  words,  it  can  be  proved  that  there  are  infinite 
systems.  That  such  demonstration  is  possible  is  affirmed 
by  Bolzano,  by  Dedekind,  by  Professor  Royce,  by  Mr. 
Russell,  and  in  fact  by  a  large  and  swelling  chorus  of 
authoritative  utterance,  scarcely  relieved  by  a  dis- 
senting voice.  After  no  little  pondering  of  the  matter, 
I  have  been  forced,  and  that,  too,  I  must  own,  against 
my  hope  and  will,  to  the  opposite  conviction.  Candour, 
then,  compels  me  to  assert,  as  I  have  elsewhere  1  briefly 
done,  not  only  that  the  arguments  which  have  been 
actually  adduced  are  all  of  them  vitiated  by  circularity, 
but  that,  in  the  very  nature  of  conception  and  inference, 
by  virtue  of  the  most  certain  standards  of  logic  itself, 
every  potential  argument,  every  possible  attempt  to 
prove  the  proposition,  is  foredoomed  to  failure,  destined 
before  its  birth  to  take  the  fatal  figure  of  the  wheel. 

The  alleged  demonstrations  are  essentially  the  same, 
being  all  of  them  but  variants  under  a  single  type.  It 
is  needless,  therefore,  in  support  of  my  first  contention, 
to  present  separate  examination  of  them  all.  Analysis 
of  one  or  two  specimens  will  suffice.  I  will  begin  with 
one  from  Bolzano's  offering,  both  because  it  marks  the 
beginning  of  the  new  era  of  thought  about  the  subject 
and  because  subsequent  writers  have  nearly  all  of  them 
either  cited  or  quoted  it,  and  that,  as  far  as  I  am  aware, 
always  with  approval.  Bolzano'  undertakes  to  demon- 

1  "The  Axiom  of  Infinity  and  Mathematical  Induction,"  BuUftin  of  tkt 
Amrruqn  Mathematical  Society,  vol.  ix.,  May,  1003. 
*  "Paradoxien,"  sect.  14. 


158  THE   AXIOM   OF   INFINITY 

strate,  among  similar  statements,  the  proposition  that 
die  Menge  der  Satze  und  Wahrheiten  an  sich  is  infinite 
(unendlicti),  this  latter  term  being  understood,  of  course, 
in  accordance  with  his  own  definition  above  given. 
The  attempt,  as  anyone  may  find  who  is  willing  to 
examine  it  minutely,  informally  postulates  as  follows: 
the  proposition,  There  are  such  truths  (as  those  con- 
templated in  the  proposition),  is  such  a  truth,  T;  T  is 
true,  is  another  such  truth,  T;  so  on;  and,  the  indicated 
process  is  inexhaustible.  Now,  these  assumptions,  which 
are  essential  to  the  argument,  and  which  any  careful 
reader  cannot  fail  to  find  implicit  in  it,  are,  possibly,  all  of 
them,  correct,  but  the  last  is  so  evident  a  petitio  principii 
as  to  make  one  look  again  and  again  lest  his  own  thought 
should  have  played  him  a  trick. 

In  case  of  Dedekind's  demonstration,  which  has  been 
heralded  far  and  wide,  the  fallacy  is  less  glaring.  The 
argument  is  far  subtler,  more  complicate,  and  the  ver- 
steckter  Zirkel  lies  deeper  in  the  folds.  But  it  is  un- 
doubtedly there,  and  its  presence  may  be  disclosed  by 
careful  explication.  Let  the  symbol  t  stand  for  thought, 
any  thought,  and  denote  by  t'  the  thought  that  t  is  a 
thought.  For  convenience,  t'  may  be  called  the  image 
of  t.  On  examination,  Dedekind's  proof  is  found  to 
postulate  as  certainties:  (i)  If  there  be  a  t,  there  is  a  /', 
image  of  /;  (2)  if  there  be  two  distinct  t's,  the  corre- 
sponding t"s  are  distinct;  (3)  there  is  a  /;  (4)  there  is  a 
t  which  is  not  a  t';  (5)  every  t  is  other  than  its  t' .  These 
being  granted,  it  is  easy  to  see,  by  supposing  each  t  to  be 
paired  with  its  /',  as  object  with  image,  that  the  assem- 
blage 6  of  all  the  t's  and  the  assemblage  6'  of  all  t"s  are 
"equivalent."  But  by  (4)  there  is  a  t  not  in  6',  which 
latter  is,  therefore,  a  part  of  6.  Hence  6  is  infinite,  by 
definition  of  the  term. 


THE  AXIOM  OF  INFINITY  159 

Let  this  matter  be  scrutinised  a  little.  Assuming 
only  the  mentioned  postulates  and,  of  course,  the  pos- 
sibility of  reflection,  it  is  obvious  that  by  pairing  the  t 
of  (4)  with  its  image  /',  then  the  latter  with  its  image, 
and  so  on,  a  sequence  5  of  /'s  is  started  which,  because 
of  (i)  and  (5),  is  incapable  of  termination.  This  S, 
too,  by  Dedekind's  proof,  is  an  infinite  assemblage. 
Accordingly,  postulate  (i),  without  which,  be  it  ob- 
served, the  proof  is  impossible,  postulates,  in  advance 
of  the  argument,  certainty  which,  if  the  argument's 
conclusion  be  true,  transcends  the  finite  before  the  infer- 
ence that  an  infinite  exists  either  is  or  can  be  drawn. 
The  reader  may  recall  how  the  Russian  mathematician 
Lobatschewsky  said,  "In  the  absence  of  proof  of  the 
Euclidian  postulate  of  parallels,  I  will  assume  that  it  is 
not  true";  and  how  thereupon  there  arose  a  new  science 
of  space.  Suppose  that,  in  like  manner,  we  say  here, 
"In  the  absence  6f  proof  that  an  act  once  found  to  be 
mentally  performable  is  endlessly  so  performable,  we 
will  assume  that  such  is  not  the  case,"  then,  whatever 
else  might  result  —  and  of  that  we  shall  presently  speak 
—  one  thing  is  at  once  absolutely  certain:  Dedekind's 
"argument"  would  be  quite  impossible.  The  fact  is 
that  a  more  beautiful  circle  than  his  is  hardly  to  be 
found  in  the  pages  of  fallacious  speculation,  or  admits 
of  construction  by  the  subtlest  instruments  of  self- 
deceiving  dialectic,  though  it  must  be  frankly  allowed 
that  Mr.  Russell's  !  more  recent  movement  about  the 
same  centre  is  equally  round  and  exquisite. 

And  this  disclosure  of  the  fatal  circle  in  the  attempted 
demonstration  serves  at  once  to  introduce  and  exemplify 
the  truth  of  my  second  contention,  which  is  that  all 
logical  discourse,  of  necessity,  ex  w  termini,  presupposes 

1  Principles  of  Mathematics,  chap,  xliii. 


l6o  THE   AXIOM   OF   INFINITY 

certainty  that  transcends  the  finite,  where  by  logical 
discourse  I  mean  such  as  consists  of  completely  deter- 
mined concepts  welded  into  a  concatenated  system  by 
the  ancient  hammer  of  deductive  logic.  The  fact  of  this 
presupposition,  of  course,  cannot  be  proved,  but,  and 
that  is  good  enough,  it  can  be  exhibited  and  beheld.  To 
attempt  to  "prove"  it  would  be  to  stultify  oneself  by 
assuming  the  possibility  of  a  deductive  argument  A 
to  prove  that  the  conclusion  of  A  cannot  be  drawn 
unless  it  is  assumed  in  advance.  The  fact,  then,  if  it 
be  a  fact,  and  of  that  there  need  not  be  the  slightest 
doubt,  is  to  be  added  to  that  small  group  of  fundamental 
simplicities  which  can  at  best  be  seen,  if  the  eye  be  fit. 

Consider,  for  example,  this  simplest  of  syllogistic 
forms:  Every  element  e  of  the  class  c  is  an  element  e 
of  the  class  c' ;  every  e  of  c'  is  an  element  e  of  the  class 
c";  .'.  every  e  of  c  is  an  e  of  c".  I  appeal  now  to  the 
reader's  own  subjective  experience  to  witness  to  the  fol- 
lowing facts:  (i)  Our  apodictic  feeling  is  the  sole  justi- 
fication of  the  inference  as  such;  (2)  that  felt  justifica- 
tion is  absolute,  neither  seeking  nor  admitting  of  appeal; 
(3)  that  sole  and  absolute  justification,  namely,  the 
apodictic  feeling,  is  in  no  slightest  degree  contingent 
upon  the  answer  to  any  question  whether  the  multitude 
of  elements  e  or  e  or  e  is  or  is  not,  may  or  may  not  be 
found  to  be,  "equivalent"  to  some  part  of  itself.  The 
feeling  of  validity  here  undoubtedly  transcends  the  finite, 
undoubtedly  holds  naught  in  reserve  against  any  possi- 
bility of  the  inference  failing  as  an  act  should  the  system 
of  elements  turn  out  to  be  infinite. 

At  some  risk  of  excessive  clearness  and  accentuation, 
for  the  matter  is  immeasurably  important,  I  venture 
to  ask  the  reader  to  witness  how  the  transcendence  or 
transfiniteness  of  certainty  shows  itself  in  yet  another 


THE  AXIOM  OF  INFINITY  l6l 

way,  not  merely  in  formal  deductive  inference,  but  also 
in  conception.  When  any  concept,  as  that  of  Parabola, 
for  example,  is  formed  or  defined,  it  is  found  that  the 
concept  contains  implicitly  a  host  of  properties  not 
given  explicitly  in  the  definition.  Properly  speaking,  the 
thing  defined  is  a  certain  organic  assemblage  of  proper- 
ties, of  which  the  totality  is  implied  in  a  properly  se- 
lected few  of  them.  Now  the  act  which  it  is  decisive 
here  to  note  is  that  by  conception  we  mean,  among 
other  things,  that  whenever  the  definition  may  present 
itself,  even  though  it  may  be  endlessly,  a  certain  in- 
variant assemblage  of  properties  implicitly  accompanies 
the  presentation.  Without  such  transfinite  certainty 
of  such  invariant  uncontingent  implication,  conception 
would  be  devoid  of  its  meaning. 

The  upshot,  then,  is  this:  that  conception  and  logical 
inference  alike  presuppose  absolute  certainty  that  an 
act  which  the  mind  finds  itself  capable  of  performing  is 
intrinsically  performable  endlessly,  or,  what  is  the  same 
thing,  that  the  assemblage  of  possible  repetitions  of  a 
once  mentally  performable  act  is  equivalent  to  some 
proper  part  of  the  assemblage.  This  certainty  I  name 
the  Axiom  of  Infinity,  and  this  axiom  being,  as  seen, 
a  necessary  presupposition  of  both  conception  and 
deductive  inference,  every  attempt  to  "demonstrate" 
the  existence  of  the  infinite  is  a  predestined  begging  of 
the  issue. 

What  follows?  Do  we,  then,  know  by  axiom  that  the 
infinite  is?  That  depends  upon  your  metaphysic.  If 
you  are  a  radical  a-priorist,  yes;  if  not,  no.  If  the  latter, 
and  I  am  now  speaking  as  an  a-priorist,  then  you  are 
agnostic  in  the  deepest  sense,  being  capable,  in  utmost 
rigour,  of  the  terms,  of  neither  conceiving  nor  inferring. 
But  if  we  do  not  know  the  axiom  to  be  true,  and  so 


162  THE  AXIOM  OF   INFINITY 

cannot  deductively  prove  the  existence  of  the  infinite, 
what,  then,  is  the  probability  of  such  existence?  The 
highest  yet  attained.  Why?  Because  the  inductive  test 
of  the  axiom,  regarded  now  as  a  hypothesis,  is  trying  to 
conceive  and  trying  to  infer,  and  this  experiment,  which 
has  been  world-wide  for  aeons,  has  seemed  to  succeed 
in  countless  cases,  and  to  fail  in  none  not  explainable  on 
grounds  consistent  with  the  retention  of  the  hypothesis. 
Finally,  to  make  briefest  application  to  a  single  con- 
crete case.  Do  the  stars  constitute  an  infinite  mul- 
titude? No  one  knows.  If  the  number  be  finite,  that 
fact  may  some  time  be  ascertained  by  actual  enumera- 
tion, and,  if  and  only  if  there  be  infinite  ensembles  of 
possible  repetitions  of  mental  processes,  it  may  also  be 
known  by  proof.  But  if  the  multitude  of  stars  be  in- 
finite, that  can  never  be  known  except  by  proof;  this 
last  is  possible  only  if  the  axiom  of  infinity  be  true,  and 
even  if  this  be  true,  the  actual  proof  may  never  be 
achieved. 


THE  PERMANENT  BASIS  OF  A  LIBERAL 
EDUCATION  l 

Is  it  possible  to  find  a  principle  or  a  set  of  principles 
qualified  to  serve  as  a  permanent  basis  for  a  theory  of 
liberal  education?  If  so,  what  is  the  principle  or  set 
of  principles?  These  are  old  questions.  We  are  living 
in  a  time  when  they  must  be  considered  anew. 

If  our  world  were  a  static  affair,  if  our  environment, 
physical,  spiritual  and  institutional,  were  stable,  then 
we  should  none  of  us  have  difficulty  in  agreeing  that  a 
liberal  education  would  be  one  that  gave  the  student 
adjustment  and  orientation  in  the  world  through  dis- 
ciplining his  faculties  in  their  relation  to  its  cardinal 
static  facts.  Such  a  world  could  be  counted  upon. 
No  one  doubts  that  in  such  a  world  it  would  be  possible 
to  find  a  permanent  basis  for  a  theory  of  liberal  edu- 
cation —  a  principle  or  a  set  of  principles  that  would 
be  adequate  and  sound,  not  merely  to-day,  but  to-day, 
yesterday  and  to-morrow. 

But  we  are  reminded  by  certain  rather  numerous 
educational  philosophers  that  our  world  is  not  a  static 
affair.  We  are  told  that  it  is  a  scene  of  perpetual 
change,  of  endless  and  universal  transformation  —  phys- 
ical flux,  institutional  flux,  social  flux,  spiritual  flux:  all 
is  flux.  These  philosophers  tell  us  of  the  rapid  and 
continued  advancement  and  multiplication  of  knowledge. 
They,  do  not  cease  to  remind  us  that  knowledge  goes 

1  Printed  in  The  Columbia  Umaersity  Quarterly,  June,  1916. 


1 64         PERMANENT   BASIS   OP   LIBERAL  EDUCATION 

on  building  itself  out,  not  only  in  all  the  old  directions, 
but  also  in  an  endlessly  increasing  variety  of  new  direc- 
tions. They  remind  us  that  the  ever-augmenting  volume 
of  knowledge  is  continually  breaking  up  into  new  divi- 
sions or  kinds,  and  that  each  of  these  quickly  asserts, 
and  sooner  or  later  demonstrates,  its  parity  with  any 
other  division  in  respect  of  utility  and  dignity  and  dis- 
ciplinary value.  They  remind  us  that  a  striking  con- 
comitant phenomenon,  which  is  partly  the  effect  of  the 
multiplication  and  differentiation  of  knowledge,  partly 
a  cause  of  it  and  partly  owing  to  other  agencies  and 
influences,  is  the  fact  that  new  occupations  constantly 
spring  into  being  on  every  hand  and  that  the  needs, 
the  desires  and  the  habits  of  men,  and  therewith  the 
drifts  and  forms  of  social  and  institutional  life,  suffer 
perpetual  mutation.  Nothing,  they  tell  us,  is  perma- 
nent except  change  itself.  All  things,  material,  mental, 
moral,  social,  institutional,  are  tossed  in  an  infinite  and 
endless  welter  of  transformations  —  evolution,  involu- 
tion, revolution,  all  going  on  at  once  and  forever. 

It  is  evident,  we  are  assured,  that  in  such  a  world 
the  search  for  abiding  principles  is  vain,  whether  we 
seek  a  permanent  basis  for  a  liberal  education  or  a  per- 
manent basis  for  anything  else.  The  doctrine  is  that  in 
our  world  permanent  bases  do  not  exist.  Permanence, 
stability,  invariance,  immutability,  there  is  none.  It 
exists  only  in  rationalistic  dreams.  It  exists  only  in 
the  insubstantial  musings  of  the  tender-minded.  It 
exists  only  in  the  cravings  of  such  as  have  not  the  prag- 
matistic  courage  or  constitution  to  deal  with  reality 
as  it  is  in  the  welter  and  the  raw.  We  are  told  that 
there  is  in  matters  educational  no  such  thing  as  eternal 
wisdom.  Wisdom  is  at  best  a  transitory  thing,  depend- 
ing on  time  and  place,  and  constantly  changing  with 


PERMANENT   BASIS   OP    LIBERAL   EDUCATION          1 6$ 

them.  A  prescription  that  is  wise  to-day  will  be  foolish 
to-morrow.  What  was  a  liberal  education  is  not  such 
now.  What  is  a  liberal  education  to-day  will  not  be 
liberal  in  the  future.  Greek  has  gone,  theology  is  gone, 
religion  is  gone,  Latin  is  almost  gone,  mathematics, 
we  are  told,  is  going,  and  so  on  and  on.  Each  branch 
of  knowledge  will  have  its  day,  and  then  will  cease  to 
be  essential.  Liberal  curricula,  it  is  contended,  must 
change  with  the  times. 

This  doctrine,  logically  conceived  and  carried  out, 
means  that  as  the  years  and  generations  follow  endlessly, 
time  and  change  will  beget  an  endless  succession  of 
so-called  liberal  curricula.  It  means  that,  if,  in  this  un- 
ending sequence,  we  observe  a  finite  number  of  success- 
ive curricula,  these  will  indeed  be  found  to  resemble  each 
other,  overlapping,  interpenetrating,  and  thus  seeming 
to  be  held  together  in  a  kind  of  unity  by  a  more  or 
less  vague  and  elusive  bond;  but  that  this  must  be 
appearance  only.  For  if  the  observed  succession  be 
prolonged,  as  it  is  bound  to  be,  the  seeming  principle 
of  unity  must  become  dimmer  and  dimmer;  the  terms 
or  curricula  of  the  endless  succession  of  them  can  have, 
in  fact,  nothing  in  common,  no  lien,  no  unity  whatever, 
save  that  pale  variety  which  serves  merely  to  constitute 
the  succession  of  curricula  an  infinite  series  of  terms. 
It  is  not  unlikely  that  the  educational  philosophers  in 
question  may  not  be  aware  that  this  is  what  their 
doctrine  means.  Nevertheless,  that  is  what  it  does 
mean. 

Is  the  doctrine  sound?  To  me  it  seems  not  to  be  so. 
The  question  is  a  question  of  fact.  The  denial  of  per- 
manent principle  and  the  assertion  of  its  concomitant 
theory  of  education  seek  to  justify  themselves  by  point- 
ing to  the  fluctuance  of  the  world.  I  do  not  deny  the 


1 66         PERMANENT   BASIS   OF   LIBERAL  EDUCATION 

fluctuance  of  the  world.  One  must  be  blind  to  do  that. 
Here,  there  and  yonder,  in  the  world  of  matter,  in  the 
world  of  mind,  in  thought,  in  religion,  in  morals,  in 
conventions,  in  institutions,  everywhere  are  evident 
the  drif tings  and  shif tings  of  events:  everywhere 
course  the  hasting  streams  of  change.  I  admit  the 
storm  and  stress,  the  tumult  and  hurly-burly  of  it  all. 
I  do  not  deny  that  impermanence  is  a  permanent  and 
mighty  fact  in  our  world.  What  I  do  deny  is  that 
impermanence  is  universal.  Its  sweep  is  not  clean. 
Far  from  it.  If  it  is,  man  has  indeed  been  a  colossal 
fool,  for  the  quest  of  Constance,  the  search  for  invari- 
ance,  for  things  that  abide,  for  forms  of  reality  that 
are  eternal,  has  been  in  all  times  and  places  the  dom- 
inant concern  of  man,  uniting  his  philosophy,  his  religion, 
his  science,  his  art  and  his  jurisprudence  into  one  mani- 
fold enterprise  of  mankind.  Not  permanence  alone, 
nor  impermanence  alone,  but  the  two  together,  one 
of  them  drawing  and  the  other  driving,  it  is  these  two 
working  together  that  have  shaped  the  course  of  human 
history  and  moulded  the  form  of  its  content.  I  admit 
that  impermanence  is  more  evident  and  obtrusive 
than  permanence,  but  I  contend  that  a  philosophy  which 
finds  in  the  world  nothing  but  change  is  a  shallow  phi- 
losophy and  false.  The  instinct  that  perpetually  drives 
man  to  seek  the  fixed,  the  stable,  the  everlasting,  has 
its  root  deep  in  nature.  It  is  a  cosmic  thing.  Must  we 
say  that  this  instinct,  this  most  imperious  of  human 
cravings,  has  no  function  except  that  of  qualifying  man 
to  be  eternally  mocked?  It  cannot  be  admitted.  The 
sweep  of  mutation  is  indeed  deep  and  wide,  but  it  is 
not  universal.  It  would  be  possible,  in  a  contest  before 
a  committee  of  competent  judges,  to  show  that  tem- 
poralities are,  in  respect  of  number,  more  than  matched 


PERMANENT   BASIS  OF   LIBERAL   EDUCATION          167 

by  eternalities,  and  that,  in  respect  of  relative  impor- 
tance, changes  are  as  dancing  wavelets  on  an  infinite 
and  everlasting  sea. 

In  our  environment  there  exist  certain  great  invariant 
massive  facts  that  now  are  and  always  will  be  necessary 
and  sufficient  to  constitute  the  basis  of  a  curriculum  or 
a  theory  of  liberal  education.  These  facts  are  obvious 
and  on  that  account  they  require  to  be  pointed  out, 
just  because,  in  the  matter  of  escaping  attention,  what 
is  very  obvious  is  a  rival  of  what  is  obscure. 

What  are  these  facts?  One  of  them  is  the  fact  that 
every  human  being  has  behind  him  an  immense  human 
past,  the  past  of  mankind.  Of  course,  I  do  not  mean 
that  what  we  call  the  human  past  is  itself  a  fixed  or 
permanent  thing.  It  is  not.  It  is  a  variable,  constantly 
changing  by  virtue  of  perpetual  additions  to  it  as  the 
years  and  centuries  empty  the  volume  of  their  events 
into  that  limitless  sea.  What  is  permanent  is  the  fact 
—  it  was  so  yesterday  and  it  will  be  so  to-morrow  — 
that  behind  each  one  of  us  there  is  a  human  past  so 
immense  as  to  be  practically  infinite.  That  fact,  I  say, 
is  permanent.  It  can  be  counted  on.  It  is  as  nearly 
eternal  as  the  race  of  man.  Out  of  that  past  we  have 
come.  Into  it  we  are  constantly  passing.  Meanwhile, 
it  is  of  the  utmost  importance  to  our  lives.  It  contains 
the  roots  of  all  we  are,  and  of  all  we  have  of  wisdom, 
of  science,  of  philosophy,  of  art,  of  jurisprudence,  of 
customs  and  institutions.  It  contains  the  record  or 
ruins  of  all  the  experiments  that  man  has  made  during 
a  quarter  or  a  half  million  years  in  the  art  of  living  in 
this  world.  This  great  stable  fact  of  an  immense  human 
past  behind  every  human  being  that  now  is  or  is  to  be, 
obviously  makes  it  necessary  for  any  theory  of  liberal 
education  to  provide  for  discipline  in  human  history 


1 68         PERMANENT   BASIS   OF   LIBERAL   EDUCATION 

and  in  the  literature  of  antiquity.  How  much?  A 
reasonable  amount  —  enough,  that  is,  to  orient  the  stu- 
dent in  relation  to  the  past,  to  give  him  a  fair  sense  of 
the  continuity  of  the  life  of  mankind,  a  decent  appre- 
ciation of  ancient  works  of  genius,  and  sense  and  knowl- 
edge enough  to  guide  his  energies  and  to  control  his 
enthusiasms  in  the  light  of  human  experience.  As  the 
centuries  go  by,  ancient  literature  and  human  history 
will  increase  more  and  more.  What  is  a  reasonable 
prescription  will,  therefore,  become  less  and  less  in  its 
relation  to  the  increasing  whole,  but  it  will  never  vanish. 
It  will  never  cease  to  be  indispensable. 

In  this  connection,  the  following  question  is  certain 
to  be  asked.  From  the  point  of  view  of  this  inquiry, 
which  aims  at  indicating  an  enduring  basis  for  a  theory 
of  liberal  education,  does  it  follow  that  Greek  or  Latin 
or  any  other  language  that  may  be  destined  to  become 
"classic"  and  "dead"  at  some  remote  future  time,  — 
does  it  follow  that  these  or  any  of  them  must  enter 
as  essential  into  the  curriculum  of  a  liberal  education? 
It  does  not.  It  would  indeed  be  a  grave  misfortune  if 
there  should  ever  come  a  time  when  there  were  no  longer 
a  goodly  number  of  scholars  devoted  to  the  great  lan- 
guages of  antiquity.  Some  of  the  thought,  of  the  sci- 
ence, of  the  wisdom,  of  the  beauty  originally  expressed 
in  these  tongues,  is,  we  have  said,  essential;  but  it  is 
precisely  the  chief  function  of  those  who  master  the 
ancient  languages  to  make  their  precious  content  avail- 
able, through  translations  and  critical  commentaries, 
for  the  great  body  of  their  fellow  men  to  whom  the  lan- 
guages themselves  must  remain  unknown.  It  is  not 
denied  that  the  scholars  in  question  will  know  and 
appreciate  such  content  as  no  others  can,  but  neither 
will  these  scholars  continue  forever  to  deny  the  possi- 


PERMANENT  BASIS  OP  LIBERAL  EDUCATION         169 

bility  of  rendering  most  of  the  content  reasonably  well 
in  the  living  languages  of  their  fellow  men.  The  con- 
trary cannot  be  much  longer  maintained.  Indeed  the 
layman  already  knows  that  Euclid,  Plato,  Aristotle, 
Aeschylus,  Sophocles,  Euripides,  Demosthenes,  Virgil, 
Cicero,  Lucretius,  and  many  others,  have  already 
learned,  or  are  rapidly  learning,  to  speak,  beautifully 
and  powerfully,  all  the  culture  languages  of  the  modern 
world. 

Another  of  the  massive  facts  that  transcends  the  flux 
of  the  world,  and  that,  therefore,  must  contribute  basic- 
ally to  any  permanent  theory  of  liberal  education, 
is  the  fact  that  every  human  being  is  encompassed  by 
a  physical  or  material  universe.  Again  I  do  not  mean, 
of  course  I  do  not  mean,  that  the  universe  remains 
always  the  same.  What  is  permanent  is  the  fact  that 
human  beings  always  have  been,  now  are,  and  always 
will  be,  surrounded  on  every  hand  by  an  infinite  objec- 
tive world  of  matter  and  force.  In  that  world  we  are 
literally  immersed.  Our  bodies  are  parts  of  it;  they 
are  composed  of  its  elements  and  will  be  resolved  into 
them  again.  If  our  minds,  too,  be  not  part  of  it,  they 
must  at  all  events,  on  pain  of  our  physical  incompe- 
tence or  extinction,  gain  and  maintain  continuous  and 
intelligent  relations  with  it.  The  great  fact  in  question, 
like  the  fact  of  the  human  past,  can  be  counted  on. 
It  survives  all  vicissitudes.  The  immersing  universe 
may  be  a  chaos  or  a  cosmos,  or  partly  chaotic  and  partly 
cosmic,  preserving  its  character  in  that  regard  or  tending 
along  an  asymptotic  path  to  chaos  complete  or  to 
cosmic  perfection.  But  if  it  is  chaotic,  we  humans 
sufficiently  match  it  in  that  regard  to  be  able  to  treat 
it  more  and  more  successfully  as  if  it  were  an  infinite 
locus  of  order  and  law.  And  we  know  that  to  do  this 


1 70         PERMANENT   BASIS   OF   LIBERAL  EDUCATION 

is  immensely  advantageous.  In  a  strict  sense,  it  is 
absolutely  indispensable.  Merely  to  live,  it  is  necessary 
to  treat  nature  as  having  some  order. 

These  considerations  show  that  any  theory  which 
aims  to  orient  and  discipline  the  faculties  of  men  and 
women  in  their  relation  to  the  great  permanent  facts 
of  the  world  must  make  basal  provision  for  discipline 
in  what  we  call  natural,  or  physical,  science.  Again, 
how  much?  Again  the  answer  is,  a  reasonable  amount. 
But  how  much,  pray,  is  that?  Enough  to  give  the 
student  a  fair  acquaintance  with  the  heroes  of  natural 
science,  a  fair  understanding  of  what  scientific  men  mean 
by  natural  order  or  law,  a  decent  insight  into  scientific 
method,  the  role  of  hypothesis,  and  the  processes  of 
experimentation  and  verification.  But  there  are  so 
many  branches  of  natural  science  and  their  number  is 
increasing.  A  liberal  curriculum  cannot  require  them 
all.  Which  shall  be  chosen?  It  does  not  matter  much. 
These  branches  differ  a  good  deal  in  content  and  in  a 
less  degree  in  method,  but  they  have  enough  in  common 
to  make  a  claim  of  superiority  for  any  one  of  them 
mainly  a  partisan  claim.  The  spirit  of  science,  its 
methods,  some  of  its  chief  results,  these  are  the  essen- 
tials. To  give  these,  physics  is  competent,  so  is  chem- 
istry, so  is  botany,  so  is  zoology,  and  so  on.  The  choice 
is  a  temporal  detail,  but  the  principle  requiring  the 
choice  is  everlasting.  A  hundred  or  a  thousand  years 
hence,  there  will  be  other  details  to  choose  from  —  sci- 
entific branches  not  yet  named,  nor  even  dreamed  of. 
But  —  and  this  is  the  point  —  a  theory  of  liberal  edu- 
cation will  not  cease  to  demand  some  discipline  in 
natural  science  so  long  as  human  beings  are  immersed 
in  an  infinite  world  of  matter  and  force. 

Nor  will  such  a  theory  fail  to  take  account  funda- 


PERMANENT   BASIS   OF   LIBERAL  EDUCATION          171 

mentally  of  a  third  great  fact  that  persists  despite  the 
flux  of  things  and  the  law  of  death.  I  refer  to  the 
fact  that  every  human  being's  fortune  depends  vitally 
upon  what  may  be  called  the  world  of  ideas.  It  is 
evident  that  of  the  total  environment  of  man,  the 
human  Gedankenwelt  is  a  stupendous  and  mighty  com- 
ponent. Like  the  other  great  components  already 
named,  or  namable,  the  world  of  ideas  is,  in  respect  of 
its  existence,  a  permanent  datum  amid  the  weltering 
sea  of  change.  Not  only  may  it  be  counted  on,  but 
it  must  be  reckoned  with.  Some  thinking  everyone  must 
do.  The  formation  and  combination  of  ideas  is  not 
merely  indispensable  to  welfare,  it  is  more  fundamental 
than  that:  it  is  essential  to  human  life.  The  world  of 
ideas  contains  countless  possibilities  that  are  not  actual- 
ized or  realized  or  validated  or  incarnated,  as  we  say, 
in  the  order  of  the  material  world,  nor  in  any  existing 
social  or  institutional  order.  It  is  plain  that  discipline 
in  the  ways  and  forms  of  abstract  thinking,  of  dealing 
with  ideas  as  ideas,  is  essential  to  a  liberal  education, 
not  merely  because  the  world  of  ideas  is  itself  a  thing 
of  supreme  and  eternal  worth,  but  because  those  who 
are  incapable  of  constructing  ideal  orders  may  not  hope 
to  have  the  imagination  requisite  for  ascertaining  or 
for  appreciating  the  frame  and  order  actualized  in  ex- 
ternal nature.  From  all  of  this  it  is  clear  that  any 
enduring  theory  of  liberal  education  must  provide  for 
the  discipline  of  logic  and  mathematics,  for  it  is  in 
these  and  these  alone  that  rigorous  or  cogent  thinking 
finds  its  standard  and  its  realization.  It  is  true  that 
most  of  the  thinking  that  the  exigencies  of  life  compel 
us  to  do  is  not  cogent  thinking.  We  are  obliged  con- 
stantly to  deal  with  ideas  that  are  too  nebulous  to 
admit  of  rigorously  logical  handling.  But  to  argue 


172          PERMANENT   BASIS   OF   LIBERAL  EDUCATION 

that  consequently  discipline  in  rigorous  thinking  is  not 
essential,  is  stupid.  It  is  to  ignore  the  value  of  stand- 
ards and  ideals.  It  is,  in  other  words,  to  be  spiritually 
blind.  I  am  making  no  partisan  plea  for  my  own  sub- 
ject. Mathematics  happens  to  be  the  name  that  time 
has  given  to  rigorous  or  cogent  thinking,  and  so  it 
happens  that  mathematics  is  the  name  of  the  one  art 
or  science  that  is  qualified  to  give  men  and  women  a 
perfect  standard  of  thinking  and  to  bring  them  into  the 
thrilling  presence  of  indestructible  bodies  of  thought. 
Call  the  science  by  any  other  name  — •  anathematics 
or  logostetics.  The  thing  itself  and  its  functions  would 
be  the  same. 

Another  cardinal  fact  among  the  permanent  consid- 
erations that  a  theory  of  liberal  education  must  rest 
upon  is  the  fact  that  human  beings  are  social  beings. 
It  is  only  in  dreams  and  romances  that  a  human  being 
lives  apart  in  isolation.  Men,  said  Aristotle,  are  made 
for  co-operation.  Every  man  and  every  woman  is  a 
born  member  of  a  thousand  teams.  Not  one  is  pure 
individual.  Each  one  is  many.  None  can  extricate 
himself  from  the  generic  web  of  man.  This  fact  sur- 
vives the  flux.  It  is  as  nearly  everlasting  as  the  human 
race.  It  is  a  rock  to  build  upon.  And  so  it  was  true 
yesterday,  is  true  to-day,  and  will  be  true  to-morrow, 
that  an  education  whose  function  it  is  to  discipline  the 
faculties  of  man  in  their  relation  to  the  great  abiding 
facts  of  life  and  the  world,  must  provide  for  discipline 
in  the  fundamentals  of  political  science.  Moreover, 
as  it  is  essential  to  the  health  and  to  the  effectiveness 
of  the  individual,  and  also  essential  to  the  welfare  of 
society  that  men  and  women  be  able  to  express  them- 
selves acceptably  and  effectively,  a  liberal  education 
will  provide  for  discipline  in  the  greatest  of  all  the  arts 


PERMANENT   BASIS   OF    LIBERAL   EDUCATION          173 

-  the  art  of  rhetoric.  No  term  has  been  more  abused, 
especially  by  amorphous  men  of  science.  Yet  the  late 
Henri  Poincar6  was  made  a  member  of  the  French 
Academy,  not  because  he  was  a  great  mathematician, 
astronomer,  physicist  and  philosopher,  but  because  of 
his  masterful  control  of  the  resources  of  the  French 
language  as  an  instrument  of  human  expression. 

I  have  spoken  of  the  invariant  fact  of  the  human  past. 
Its  complement  is  the  fact  of  the  human  future.  That, 
too,  is  a  great  abiding  fact.  It  is,  in  practice,  to  be 
treated  as  eternal,  for,  if  the  race  of  man  be  doomed 
to  extinction,  then,  in  that  far  off  event,  human  edu- 
cation itself  will  cease.  Does  it  follow  that  a  theory  of 
liberal  education  must  provide  for  instruction  in  proph- 
ecy? It  does  follow.  But  is  it  not  foolish  to  speak 
of  instruction  in  prophecy?  For  is  not  prophecy  a 
thing  of  the  past?  Is  it  not  a  dead  or  a  dying  office 
of  priests?  It  is  not  foolish,  it  is  not  a  thing  of  the 
past,  it  is  not  a  dead  or  dying  office  of  priests.  Proph- 
ecy is  a  thing  of  the  present,  destined  to  increase  with 
the  advancement  of  knowledge.  Every  department  of 
study  is  a  department  of  prophecy.  It  is  the  function 
of  science  to  foretell.  Prophecy  is  not  the  opposite  of 
history,  it  is  history's  main  function.  As  W.  K.  Clif- 
ford long  ago  pointed  out,  every  proposition  in  physics 
or  astronomy  or  chemistry  or  zoology  or  mathematics, 
or  other  branch  of  science,  is  a  rule  of  conduct  facing 
the  future  —  a  rule  saying  that,  if  such-and-such  be 
true,  then  such-and-such  must  be  true;  if  such-and-such 
a  situation  be  present,  then  such-and-such  things  will 
happen;  if  we  do  thus-and-thus,  then  certain  statable 
consequences  may  be  expected.  Foretelling,  indeed, 
is  not  the  exclusive  office  of  knowledge,  for  musing, 
meditation,  pensiveness,  pure  contemplation,  have  their 


174         PERMANENT   BASIS   OF   LIBERAL  EDUCATION 

legitimate  place;  but  man  is  mainly  and  primarily  an 
active  animal;  and  in  relation  to  action,  the  business 
of  knowledge  is  prophecy,  forecasting  what  to  do  and 
what  to  expect. 

Finally  it  remains  to  mention  another  fundamental 
matter  that  must  contribute  in  a  paramount  measure 
to  any  just  theory  of  a  liberal  education.  It  is  not  a 
matter  strictly  co-ordinate  with  the  other  matters 
mentioned,  but  it  touches  them  all,  penetrates  them 
all  and  transfigures  them  all.  I  refer  to  the  discipline 
of  beauty.  Beauty  is  the  most  vitalizing  thing  in  the 
world.  It  is  beauty  that  makes  life  worth  living  and 
makes  it  possible.  If,  by  some  fiendish  cataclysm, 
all  the  beauty  of  art  and  all  the  beauty  of  nature  were 
to  be  suddenly  blotted  out,  the  human  race  would 
quickly  perish  through  depression  caused  by  the  ubiq- 
uitous presence  of  ugliness.  Does  it  follow  that  a 
liberal  curriculum  must  provide  for  the  instruction  of 
every  student  in  all  the  arts?  No.  Like  the  natural 
sciences,  the  arts  are  enough  alike  to  make  any  one  of 
them  a  representative  of  them  all.  Besides,  all  sub- 
jects of  study  are  penetrated  with  beauty,  and  any 
one  of  them  may  be  so  administered  as  to  enlarge 
and  refine  the  sense  of  what  is  beautiful  in  life  and 
the  world. 

Such  I  take  to  be  major  considerations  among  the 
great  permanent  massive  facts  that  together  suffice 
and  are  essential  to  constitute  an  enduring  basis  for  a 
theory  of  liberal  education.  Ought  discipline  to  be 
prescribed  in  all  the  indicated  fields?  The  answer 
would  seem  to  be  that  a  liberally  educated  man  or 
woman  is  one  who  has  been  instructed  in  them  all.  It 
follows  that  there  be  seekers  who  are  by  nature  not 
qualified  to  find.  But  in  the  case  of  these,  as  in  the 


PERMANENT  BASIS  OF  LIBERAL  EDUCATION         175 

case  of  their  more  gifted  fellows,  it  must  be  remem- 
bered that  not  the  least  service  a  program  of  liberal 
study  should  render,  is  that  of  disclosing  to  men  and 
women  and  to  their  fellows  their  respective  powers 
and  limitations. 


GRADUATE  MATHEMATICAL  INSTRUCTION  FOR 
GRADUATE  STUDENTS  NOT  INTENDING  TO 
BECOME  MATHEMATICIANS1 

IN  his  "Annual  Report"  under  date  of  November 
last,  the  President  of  Columbia  University  speaks  in 
vigorous  terms  of  what  he  believes  to  be  the  increasing 
failure  of  present-day  advanced  instruction  to  fulfil  one 
of  the  chief  purposes  for  which  institutions  of  higher 
learning  are  established  and  maintained. 

In  the  course  of  an  interesting  section  devoted  to 
college  and  university  teaching,  President  Butler  says: 

A  matter  that  is  closely  related  to  poor  teaching  is  found  in  the  grow- 
ing tendency  of  colleges  and  universities  to  vocationalize  all  their  instruc- 
tion. A  given  department  will  plan  all  its  courses  of  instruction  solely 
from  the  point  of  view  of  the  student  who  is  going  to  specialize  in  that 
field.  It  is  increasingly  difficult  for  those  who  have  the  very  proper  desire 
to  gain  some  real  knowledge  of  a  given  topic  without  intending  to  become 
specialists  in  it.  A  university  department  is  not  well  organized  and  is  not 
doing  its  duty  until  it  establishes  and  maintains  at  least  one  strong  sub- 
stantial university  course  designed  primarily  for  students  of  maturity  and 
power,  which  course  will  be  an  end  in  itself  and  will  present  to  those  who 
take  it  a  general  view  of  the  subject-matter  of  a  designated  field  of  knowl- 
edge, its  methods,  its  literature  and  its  results.  It  should  be  possible  for 
an  advanced  student  specializing  in  some  other  field  to  gain  a  general 
knowledge  of  physical  problems  and  processes  without  becoming  a  physicist; 
or  a  general  knowledge  of  chemical  problems  and  processes  without  becoming 
a  chemist;  or  a  general  knowledge  of  zoological  problems  and  processes 
without  becoming  a  zoologist;  or  a  general  knowledge  of  mathematical 
problems  and  processes  without  becoming  a  mathematician. 

1  An  address  delivered  before  Section  A  of  the  American  Association 
for  the  Advancement  of  Science,  December  30,  1914.  Printed  in  Science, 
March  26,  1915. 


GRADUATE   MATHEMATICAL   INSTRUCTION  177 

This  is  a  large  matter,  involving  all  the  cardinal 
divisions  of  knowledge.  I  have  neither  time  nor  com- 
petence to  deal  with  it  fully  or  explicitly  in  all  its  bear- 
ings. As  indicated  by  the  title  of  this  address  it  is  my 
intention  to  confine  myself,  not  indeed  exclusively  but 
in  the  main,  to  consideration  of  the  question  in  its 
relation  to  advanced  instruction  in  mathematics.  The 
obvious  advantages  of  this  restriction  will  not,  I  believe, 
be  counterbalanced  by  equal  disadvantages.  For,  much 
as  the  principal  subjects  of  university  instruction  differ 
among  themselves,  it  is  yet  true  that  as  instruments 
of  education  they  have  a  common  character  and  for 
their  efficacy  as  such  depend  fundamentally  upon  the 
same  educational  principles.  A  discussion,  therefore, 
of  an  important  and  representative  part  of  the  general 
question  will  naturally  derive  no  little  of  whatever 
interest  and  value  it  may  have  from  its  implicit  bearing 
upon  the  whole.  It  is  not  indeed  my  intention  to 
depend  solely  upon  such  implicit  bearings  nor  upon  the 
representative  character  of  mathematics  to  intimate  my 
opinion  respecting  the  question  in  its  relation  to  other 
subjects.  On  the  contrary,  I  am  going  to  assume  that 
specialists  in  other  fields  will  allow  me,  as  a  lay  neigh- 
bor fairly  inclined  to  minding  his  own  affairs,  the  priv- 
ilege of  some  quite  explicit  preliminary  remarks  upon 
the  larger  question. 

I  suspect  that  my  interest  in  the  matter  is  in  a  meas- 
ure temperamental;  and  my  conviction  in  the  premises, 
though  it  is  not,  I  believe,  an  unreasoned  one,  may  be 
somewhat  colored  by  inborn  predilection.  At  all  events 
I  own  that  a  good  many  years  of  devotion  to  one  field 
of  knowledge  has  not  destroyed  in  me  a  certain  fondness 
for  avocational  studies,  for  books  that  deal  with  large 
subjects  in  large  ways,  and  for  men  who,  uniting  the 


178  GRADUATE   MATHEMATICAL  INSTRUCTION 

generalist  with  the  specialist  in  a  single  gigantic  per- 
sonality, can  show  you  perspectives,  contours  and  reliefs, 
a  great  subject  or  a  great  doctrine  in  its  principal 
aspects,  in  its  continental  bearings,  without  first  com- 
pelling you  to  survey  it  pebble  by  pebble  and  inch  by 
inch.  I  can  not  remember  the  time  when  it  did  not 
seem  to  me  to  be  the  very  first  obligation  of  universities 
to  cherish  instruction  of  the  kind  that  is  given  and 
received  in  the  avocational  as  distinguished  from  the 
vocational  spirit  —  the  kind  of  instruction  that  has  for 
its  aim,  not  action  but  understanding,  not  utilities  but 
ideas,  not  efficiency  but  enlightenment,  not  prosperity 
but  magnanimity.  For  without  intelligence  and  mag- 
nanimity —  without  light  and  soul  —  no  form  of  being 
can  be  noble  and  every  species  of  conduct  is  but  a  kind 
of  blundering  in  the  night.  I  could  hardly  say  more 
explicitly  that  I  agree  heartily  and  entirely  with  the 
main  contention  of  President  Butler's  pronouncement. 
Indeed  I  should  go  a  step  further  than  he  has  gone.  He 
has  said  that  a  university  department  is  not  well  organ- 
ized and  is  not  doing  its  duty  until  it  establishes  and 
maintains  the  kind  of  instruction  I  have  tried  to  char- 
acterize. To  that  statement  I  venture  to  add  explicitly 
—  what  is  of  course  implicit  in  it  —  that  a  university  is 
not  well  organized  and  is  not  doing  its  duty  until  it 
makes  provision  whereby  the  various  departments  are 
enabled  to  foster  the  kind  of  instruction  we  are  talking 
about.  That  in  all  major  subjects  of  university  instruc- 
tion there  ought  to  be  given  courses  designed  for  stu- 
dents of  "maturity  and  power"  who,  whilst  specializing 
in  one  subject  or  one  field,  desire  to  generalize  in  others, 
appears  to  me  to  be  from  every  point  of  view  so  rea- 
sonable and  just  a  proposition  that  it  would  not  occur 
to  me  to  regard  it  as  questionable  or  debatable  were  it 


GRADUATE   MATHEMATICAL  INSTRUCTION  179 

not  for  the  fact  that  it  actually  is  questioned  and  debated 
by  teachers  of  eminence  and  authority. 

What  is  there  in  the  contention  about  which  men  may 
differ?  Dr.  Butler  has  said  that  there  is  a  "growing 
tendency  of  college  and  university  departments  to  vo- 
cationalize  all  their  instruction."  Is  the  statement 
erroneous?  It  may,  I  think,  be  questioned  whether  the 
tendency  is  growing.  I  hope  it  is  not.  Of  course 
specialization  is  not  a  new  thing  in  the  world.  It 
is  far  older  than  history.  Let  it  be  granted  that  it 
is  here  to  stay,  for  it  is  indispensable  to  the  advance- 
ment of  knowledge  and  to  the  conduct  of  human  affairs. 
Every  one  knows  that.  There  is,  however,  some 
evidence  that  specialization  is  becoming,  indeed  that  it 
has  become,  wiser,  less  exclusive,  more  temperate. 
The  symptoms  of  what  not  long  ago  promised  to  become 
a  kind  of  specialism  mania  appear  to  be  somewhat  less 
pronounced.  Recognition  of  the  fact  that  specializa- 
tion is  in  constant  peril  of  becoming  so  minute  and 
narrow  as  to  defeat  its  own  ends  is  now  a  commonplace 
among  specialists  themselves,  many  of  whom  have 
learned  the  lesson  through  sad  experience,  others  from 
observation.  Specialists  are  discoverers.  One  of  our 
recent  discoveries  is  the  discovery  of  a  very  old  truth: 
we  have  discovered  that  no  work  can  be  really  great 
which  does  not  contain  some  element  or  touch  of  the 
universal,  and  that  is  not  exactly  a  new  insight.  Leo- 
nardo da  Vinci  says: 

We  may  frankly  admit  that  certain  people  deceive  themselves  who 
apply  the  title  "a  good  master"  to  a  painter  who  can  only  do  the  head  or 
the  figure  well.  Surely  it  is  no  great  achievement  if  by  studying  one  thing 
only  during  his  whole  lifetime  he  attain  to  some  degree  of  excellence  therein ! 

The  conviction  seems  to  be  gaining  ground  that  in 
the  republic  of  learning  the  ideal  citizen  is  neither 


l8o  GRADUATE    MATHEMATICAL   INSTRUCTION 

the  ignorant  specialist,  however  profound  he  may  be, 
nor  the  shallow  generalist,  however  wide  the  range  of 
his  interest  and  enlightenment.  It  is  not  important, 
however,  in  this  connection  to  ascertain  whether  the 
vocationalizing  tendency  is  at  present  increasing  or  de- 
creasing or  stationary.  What  is  important  is  to  recog- 
nize the  fact  that  the  tendency,  be  it  waxing  or  waning, 
actually  exists,  and  that  it  operates  in  such  strength 
as  practically  to  exclude  all  provision  for  the  student 
who,  if  I  may  so  express  it,  would  qualify  himself  to 
gaze  into  the  heavens  intelligently  without  having  to 
pursue  courses  designed  for  none  but  such  as  would 
emulate  a  Newton  or  a  Laplace.  If  any  one  doubts  that 
such  is  the  actual  state  of  the  case,  the  remedy  is  very 
simple:  let  him  choose  at  random  a  dozen  or  a  score  of 
the  principal  universities  and  examine  their  bulletins 
of  instruction  in  the  major  fields  of  knowledge. 

Another  element  —  an  extremely  important  element  — 
of  President  Butler's  contention  is  present  in  the  form 
of  a  double  assumption :  it  is  assumed  that  in  any  uni- 
versity community  there  are  serious  and  capable  students 
whose  primary  aim  is  indeed  the  winning  of  mastery 
in  a  chosen  field  of  knowledge  but  who  at  the  same  time 
desire  to  gain  some  understanding  of  other  fields  — 
some  intelligence  of  their  enterprises,  their  genius,  their 
methods  and  their  achievements;  it  is  further  assumed 
that  this  non-vocational  or  avocational  propensity  is 
legitimate  and  laudable.  Are  the  assumptions  correct? 
The  latter  one  involves  a  question  of  values  and  will 
be  dealt  with  presently.  In  respect  of  the  former 
we  have  to  do  with  what  mathematicians  call  an  exist- 
ence theorem:  Do  the  students  described  exist?  They 
do.  Can  the  fact  be  demonstrated  —  deductively 
proved?  It  can  not.  How,  then,  may  we  know  it  to  be 


GRADUATE   MATHEMATICAL   INSTRUCTION  l8l 

true?  The  answer  is:  partly  by  observation,  partly 
by  experience,  partly  by  inference  and  partly  by  being 
candid  with  ourselves.  Who  is  there  among  us  that  is 
unwilling  to  admit  that  he  himself  now  is  or  at  least 
once  was  a  student  of  the  kind?  Where  is  the  univer- 
sity professor  to  whom  such  students  have  not  revealed 
themselves  as  such  in  conversation?  Who  is  it  that  has 
not  learned  of  their  existence  through  the  testimony  of 
others?  No  doubt  some  of  us  not  only  have  known 
students  of  the  kind,  but  have  tried  in  a  measure  to 
serve  them.  We  may  as  well  be  frank.  I  have  myself 
for  some  years  offered  in  my  subject  a  course  designed 
in  large  part  for  students  having  no  vocational  interest 
in  mathematics.  I  may  be  permitted  to  say,  for  what 
the  testimony  may  be  worth,  that  the  response  has  been 
good.  The  attendance  has  been  composed  about  equally 
of  students  who  were  not  looking  forward  to  a  career 
in  mathematics  and  of  students  who  were.  And  this 
leads  me  to  say,  in  passing,  that,  if  the  latter  students 
were  asked  to  explain  what  value  such  instruction  could 
have  for  them,  they  would  probably  answer  that  it 
served  to  give  them  some  knowledge  about  a  great  sub- 
ject which  they  could  hardly  hope  to  acquire  from 
courses  designed  solely  to  give  knowledge  of  the  subject. 
Every  one  knows  that  it  often  is  of  great  advantage  to 
treat  a  subject  as  an  object.  One  of  the  chief  values 
of  //-dimensional  geometry  is  that  it  enables  us  to  con- 
template ordinary  space  from  the  outside,  as  even  those 
who  have  but  little  imagination  can  contemplate  a 
plane  because  it  does  not  immerse  them.  Returning 
from  this  digression,  permit  me  to  ask:  if,  without 
trying  to  discover  the  type  of  student  in  question,  we 
yet  become  aware,  quite  casually,  that  the  type  actually 
exists,  is  it  not  legitimate  to  infer  that  it  is  much  more 


1 82  GRADUATE   MATHEMATICAL  INSTRUCTION 

numerously  represented  than  is  commonly  supposed? 
And  if  such  students  occasionally  make  their  presence 
known  even  when  we  do  not  offer  them  the  kind  of 
instruction  to  render  their  wants  articulate,  is  it  not 
reasonable  to  infer  that  the  provision  of  such  instruction 
would  have  the  effect  of  revealing  them  in  much  greater 
numbers? 

Indeed  it  does  not  seem  unreasonable  to  suppose 
that  a  "strong  substantial  course"  of  the  kind  in  ques- 
tion, in  whatever  great  subject  it  were  given,  would  be 
attended  not  only  by  considerable  numbers  of  regular 
students  but  in  a  measure  also  by  officers  of  instruction 
in  other  subjects  and  even  perhaps  by  other  qualified 
residents  of  an  academic  community.  Only  the  other 
day  one  of  my  mathematical  colleagues  said  to  me 
that  he  would  rejoice  in  an  opportunity  to  attend  such 
a  course  in  physics.  The  dean  of  a  great  school  of  law 
not  long  ago  expressed  the  wish  that  some  one  might 
write  a  book  on  mathematics  in  such  a  way  as  would 
enable  students  like  himself  to  learn  something  of  the 
innerness  of  this  science,  something  of  its  spirit,  its 
range,  its  ways,  achievements  and  aspiration.  I  have 
known  an  eminent  professor  of  economics  to  join  a 
beginner's  class  in  analytical  geometry.  Very  recently 
one  of  the  major  prophets  of  philosophy  declared  it  to 
be  his  intention  to  suspend  for  a  season  his  own  special 
activity  in  order  to  devote  himself  to  acquiring  some 
knowledge  of  modern  mathematics.  Similar  instances 
abound  and  might  be  cited  by  any  one  not  only  at 
great  length,  but  in  connection  with  every  cardinal 
division  of  knowledge.  Their  significance  is  plain. 
They  are  but  additional  tokens  of  the  fact  that  the 
race  of  catholic-minded  men  has  not  been  extinguished 
by  the  reigning  specialism  of  the  time,  but  that  among 


GRADUATE  MATHEMATICAL  INSTRUCTION  183 

students  and  scholars  there  are  still  to  be  found  those 
whose  curiosity  and  intellectual  interests  surpass  all 
professional  limits  and  crave  instruction  more  generic 
in  kind,  more  liberal,  if  you  please,  and  ampler  in  its 
scope,  than  our  vocationalized  programs  afford. 

As  to  the  question  of  values,  I  maintain  that  the  desire 
of  such  men  is  entirely  legitimate,  that  it  is  wholesome 
and  praiseworthy,  that  it  deserves  to  be  stimulated, 
and  that  universities  ought  to  meet  it,  if  they  can. 
Indeed,  all  this  seems  to  me  so  obvious  that  I  find  it  a 
little  difficult  to  treat  it  seriously  as  a  question.  If  the 
matter  must  be  debated,  let  it  be  debated  on  worthy 
ground.  To  say,  as  proponents  sometimes  say,  that, 
inasmuch  as  all  knowledge  turns  out  sooner  or  later  to 
be  useful,  students  preparing  for  a  given  vocation  by 
specializing  in  a  given  field  may  profitably  seek  some 
general  acquaintance  with  other  fields  because  such 
general  knowledge  will  indirectly  increase  their  voca- 
tional equipment,  is  to  offer  a  consideration  which, 
though  in  itself  it  is  just  enough,  yet  degrades  the  dis- 
cussion from  its  appropriate  level,  which  is  that  of  an 
ideal  humanity,  down  to  the  level  of  mere  efficiency  and 
practicianism.  No  doubt  one  engaged  in  minutely 
studying  the  topography  of  a  given  locality  because  he 
intends  to  reside  in  it  might  be  plausibly  advised  to 
study  also  the  general  geography  of  the  globe  on  the 
ground  that  his  special  topographical  knowledge  would 
be  thus  enhanced,  and  that,  moreover,  he  might  some 
time  desire  to  travel.  But  if  we  ventured  to  counsel 
him  so,  he  might  reply:  What  you  say  is  true.  But 
why  do  you  ply  me  with  such  low  considerations?  Why 
do  you  regard  me  as  something  crawling  on  its  belly? 
Don't  you  know  that  I  ought  to  acquire  a  general  knowl- 
edge, of  geography,  not  primarily  because  it  may  be 


184  GRADUATE   MATHEMATICAL  INSTRUCTION 

useful  to  me  as  a  resident  here  or  as  a  possible  traveler, 
but  because  such  knowledge  is  essential  to  me  in  my 
character  as  a  man?  The  rebuke,  if  we  were  fortunately 
capable  of  feeling  it,  would  be  well  deserved.  A  man 
building  a  bridge  is  greater  than  the  engineer;  a  man 
planting  seed  is  greater  than  the  farmer;  a  man  teach- 
ing calculus  is  greater  than  the  mathematician;  a  man 
presiding  at  a  faculty  meeting  is  greater  than  the  dean 
or  the  president.  We  may  as  well  remember  that  man 
is  superior  to  any  of  his  occupations.  His  supreme 
vocation  is  not  law  nor  medicine  nor  theology  nor  com- 
merce nor  war  nor  journalism  nor  chemistry  nor  physics 
nor  mathematics  nor  literature  nor  any  specific  science 
or  art  or  activity;  it  is  intelligence,  and  it  is  this  supreme 
vocation  of  man  as  man  that  gives  to  universities 
their  supreme  obligation.  It  is  unworthy  of  a  university 
to  conceive  of  man  as  if  he  were  created  to  be  the  servant 
of  utilities,  trades,  professions  and  careers:  these  things 
are  for  him:  not  ends  but  means.  It  is  said  that  intel- 
ligence is  good  because  it  prospers  us  in  our  trades, 
industries  and  professions;  it  ought  to  be  said  that  these 
things  are  good  because  and  in  so  far  as  they  prosper 
intelligence.  Even  if  we  do  not  conceive  the  office  of 
intelligence  to  be  that  of  contributing  to  being  in  its 
highest  form,  which  consists  in  understanding,  even  if 
we  conceive  its  function  less  nobly  as  that  of  enabling 
us  to  adjust  ourselves  to  our  environment,  the  same  con- 
clusion holds.  For  what  is  our  environment?  Is  it 
wholly  or  mainly  a  matter  of  sensible  circumstance  — 
sea  and  land  and  sky,  heat  and  cold,  day  and  night, 
seasons,  food,  raiment,  and  the  like?  Far  from  it.  It 
is  rather  a  matter  of  spiritual  circumstances  —  ideas, 
sentiments,  doctrines,  sciences,  institutions,  and  arts. 
It  is  in  respect  of  this  ever-changing  and  ever-devel- 


GRADUATE   MATHEMATICAL  INSTRUCTION  185 

oping  world  of  spiritual  things,  it  is  in  respect  of  this 
invisible  and  intangible  environment  of  life,  that  uni- 
versities, whilst  aiming  to  give  mastery  in  this  part  or 
that,  are  at  the  same  time  under  equal  obligation  to 
give  to  such  as  can  receive  it  some  general  orientation 
in  the  whole. 

And  now  as  to  the  question  of  feasibility.  Can  the 
thing  be  done?  So  far  as  mathematics  is  concerned  I 
am  confident  that  it  can,  and  I  have  a  strong  lay  sus- 
picion that  it  can  be  done  in  all  other  subjects. 

It  is  my  main  purpose  to  show,  with  some  regard  to 
concreteness  and  detail,  that  the  thing  is  feasible  in 
mathematics.  Before  doing  so,  however,  I  desire  to 
view  the  matter  a  little  further  in  its  general  aspect 
and  in  particular  to  deal  with  some  of  the  considera- 
tions that  tend  to  deter  many  scientific  specialists  from 
entering  upon  the  enterprise. 

One  of  the  considerations,  and  one,  too,  that  is  often 
but  little  understood,  and  so  leads  to  wrong  impu- 
tations of  motive,  though  it  is  in  a  sense  distinctly 
creditable  to  those  who  are  influenced  by  it,  is  the  con- 
sideration that  relates  to  intricacy  and  technicality  of 
subject-matter  and  doctrine.  Every  specialist  knows 
that  the  principal  developments  in  his  branch  of  science 
are  too  intricate,  too  technical  and  too  remote  from  the 
threshold  of  the  matter  to  be  accessible  to  laymen, 
whatever  their  abilities  and  attainments  in  foreign 
fields.  Not  only  does  he  know  that  there  is  thus  but 
relatively  little  of  his  science  which  laymen  can  under- 
stand but  he  knows  also  that  the  portions  which  they 
can  not  understand  are  in  general  precisely  those  of 
greatest  interest  and  beauty.  And  knowing  this,  he 
feels,  sometimes  very  strongly,  that  were  he  to  endeavor 
by  means  of  a  lecture  course  to  give  laymen  a  general 


1 86  GRADUATE   MATHEMATICAL  INSTRUCTION 

acquaintance  with  his  subject,  he  could  not  fail  to  incur 
the  guilt  of  giving  them,  not  merely  an  inadequate 
impression,  but  an  essentially  false  impression,  of  the 
nature,  significance  and  dignity  of  a  great  field  of  knowl- 
edge. His  hesitance,  therefore,  is  not  due,  as  it  is  some- 
times thought  to  be,  to  indifference  or  to  selfishness. 
Rather  is  it  due  to  a  sense  of  loyalty  to  truth,  to  a 
sense  of  veracity,  to  an  unwillingness  to  mislead  or  de- 
ceive. Of  course  strange  things  do  sometimes  happen, 
and  it  is  barely  conceivable  that  once  in  a  long  time 
nature  may,  in  a  sportive  mood,  produce  a  kind  of 
specialist  whose  subject  affects  him  much  as  the  pos- 
session of  an  apple  or  a  piece  of  candy  affects  the  boy 
who  goes  round  the  corner  in  order  to  have  it  all  him- 
self. But  if  the  type  exist,  not  many  men  could  claim 
the  odd  distinction  of  belonging  to  it.  Specialists  are 
as  generous  and  humane  as  other  men.  Their  subjects 
affect  them  as  that  same  boy  is  affected  when,  if  he 
chance  to  come  suddenly  upon  some  strange  kind  of 
flower  or  bird,  he  at  once  summons  his  sister  or  brother 
or  father  or  mother  or  other  friend  to  share  in  his 
surprise  and  joy.  There  is  this  difference,  however  — 
the  specialist  must,  unfortunately,  suffer  his  joy  in 
solitude  unless  and  until  he  finds  a  comrade  in  kind. 
I  admit  that  the  deterrent  consideration  in  question  is 
thoroughly  intelligible.  I  contend  that  the  motive  it 
involves  presents  an  attractive  aspect.  But  I  can  not 
think  it  of  sufficient  weight  to  be  decisive.  It  involves, 
I  believe,  an  erroneous  estimate  of  values,  a  fallacious 
view  of  the  ways  of  truth  to  men.  A  few  years  ago, 
when  making  a  railway  journey  through  one  of  the  most 
imposing  parts  of  the  Rocky  Mountains,  I  was  tempted 
like  many  another  passenger  to  procure  some  photo- 
graphs of  the  scenery  in  order  to  convey  to  far-away 


GRADUATE   MATHEMATICAL   INSTRUCTION  187 

friends  some  notion  of  the  wonders  of  it.  So  far, 
however,  did  the  actual  scenery  surpass  the  pictures  of 
it,  excellent  as  these  were,  that  I  decided  not  to  buy 
them,  feeling  it  were  better  to  convey  no  impression 
at  all  than  to  give  one  so  inferior  to  my  own.  No 
doubt  the  decision  might  be  defended  on  the  ground  of 
its  motive.  Did  it  not  originate  in  a  certain  laudable 
sense  of  obligation  to  truth?  Nevertheless,  as  I  am  now 
convinced,  the  decision  was  silly.  For  in  accordance 
with  the  same  principle  it  is  plain  that  I  ought  to  have 
wished  to  have  my  own  impressions  erased,  seeing  that 
they  must  have  been  quite  inferior  to  those  of  a  widely 
experienced  mountaineer  as  those  which  the  pictures 
could  have  given  were  inferior  to  mine.  Who  is  so 
foolish  as  to  argue  that  no  one  should  learn  anything 
about,  say  London,  unless  he  means  to  master  all  its 
plans,  its  architecture  and  its  history  in  their  every 
phase,  feature  and  detail?  Who  would  contend  that 
because  we  are  permitted  to  know  only  so  little  of 
what  is  happening  in  the  European  war,  we  ought  to 
remain  in  total  ignorance  of  it?  Who  would  say  that 
no  one  may  with  propriety  seek  to  learn  something 
about  ancient  Rome  unless  he  is  bent  on  becoming  a 
Gibbon  or  a  Mommsen?  It  is  undoubtedly  true  that 
an  endeavor  to  present  a  body  of  doctrine  or  a  science 
to  such  as  can  not  receive  it  fully  must  result  in  giving 
a  false  impression  of  the  truth.  But  the  notion  that 
such  an  endeavor  is  therefore  wrong  is  a  notion  which, 
if  consistently  and  thoroughly  carried  out,  would  put 
the  human  mind  entirely  out  of  commission.  All  im- 
pressions, all  views,  all  theories,  all  doctrines,  all  sciences 
are  false  in  the  sense  of  being  partial,  imperfect,  incom- 
plete. "II  n'y  a  plus  des  problemes  resolus  et  d'autres 
qui  ne  le  sont  pas,  il  y  a  seulement  des  probldmes 


1 88  GRADUATE   MATHEMATICAL   INSTRUCTION 

plus  ou  mains  resolus,"  said  Henri  Poincare.  Every 
one  must  see  that,  but  for  the  helpfulness  of  views 
which  because  incomplete  are  also  in  a  measure  false, 
even  the  practical  conduct  of  life,  not  to  say  the  advance- 
ment of  science,  would  be  impossible.  There  is  no 
other  choice:  either  we  must  subsist  upon  fragments  or 
perish. 

Again,  many  a  specialist  shrinks  from  trying  to  pre- 
sent his  subject  to  laymen  because  he  looks  upon  such 
activity  as  a  species  of  what  is  called  popularization  of 
science,  and  he  believes  that  such  popularization,  even 
in  its  best  sense,  closely  resembles  vulgarization  in  its 
worst.  He  fancies  that  there  is  a  sharp  line  bounding 
off  knowledge  that  is  mere  knowledge  from  knowledge 
that  is  scientific.  In  his  view  science  is  for  specialists 
and  for  specialists  only.  He  declines,  on  something 
like  moral  and  esthetic  grounds,  to  engage  in  what  he 
calls  playing  to  the  gallery.  It  might,  of  course,  be 
said  that  there  is  more  than  one  way  of  playing  to  the 
gallery.  It  could  be  said  that  one  way  consists  in 
acting  the  role  of  one  who  imagines  that  his  intellectual 
interests  are  so  austere  and  elevated  and  his  thought 
so  profound  that  a  just  sense  of  the  awful  dignity  of 
his  vocation  imposes  upon  him,  when  in  presence  of 
the  vulgar  multitude,  the  solemn  law  of  silence.  It 
would  be  ungenerous,  however,  if  not  unfair,  to  insist 
upon  the  justice  of  such  a  possible  retort.  Rather  let 
it  be  granted,  for  it  is  true,  that  much  so-called  popu- 
larization of  science  is  vicious,  relieving  the  ignorant  of 
their  modesty  without  relieving  them  of  their  ignorance, 
equipping  them  with  the  vocabulary  of  knowledge 
without  its  content  and  so  fostering  not  only  a  vain  and 
empty  conceit,  but  a  certain  facility  of  speech  that  is 
seemly,  impressive  and  valuable  only  when,  as  is  too 


GRADUATE   MATHEMATICAL  INSTRUCTION  189 

seldom  the  case,  it  is  accompanied  by  solid  attainments. 
To  say  this,  however,  is  not  to  lay  an  indictment  against 
that  kind  of  scientific  popularization  which  was  so 
happily  illustrated  by  the  very  greatest  men  of  antiquity, 
which  was  not  disdained  even  by  Galileo  in  the  begin- 
nings of  modern  science  nor  by  Leonardo  da  Vinci,  and 
which  in  our  own  time  has  engaged  the  interest  and 
skill  of  such  men  as  Clifford  and  Helmholtz,  Haeckel 
and  Huxley,  Mach,  Ostwald,  Enriques  and  Henri  Poin- 
car6.  It  is  not  to  arraign  that  variety  of  popularization 
which  any  one  may  behold  in  the  constant  movement  of 
ideas,  once  reserved  exclusively  for  graduate  students, 
down  into  undergraduate  curricula  and  which  has,  for 
example,  made  the  doctrine  of  limits,  analytical  geom- 
etry, projective  geometry,  and  the  notions  of  the  deriva- 
tive and  the  integral  available  for  presentation  to  college 
freshmen  or  even  to  high-school  pupils.  It  is  not  to 
condemn  that  kind  of  popularization  which  is  so  nat- 
ural a  process  that  it  actually  goes  on  in  a  thousand 
ways  all  about  us  without  our  deliberate  cooperation, 
without  our  intention  or  our  consent,  and  has  enriched 
the  common  sense  and  common  knowledge  of  our  time 
with  countless  precious  elements  from  among  the  sci- 
entific and  philosophic  discoveries  made  by  other 
generations  of  men. 

Finally  it  remains  to  mention  the  important  type  of 
specialist  in  whom  strongly  predominates  the  predilection 
for  research  as  distinguished  from  exposition.  He  knows, 
as  every  one  knows,  that  through  what  is  called  practical 
applications  of  science  many  a  scientific  discovery  is 
made  to  serve  innumerable  human  beings  who  do  not 
understand  it  and  innumerable  others  who  never  can. 
He  may  or  may  not  believe  in  a  vocational  instruction; 
he  may  or  may  not  regard  intelligence  as  an  ultimate 


GRADUATE   MATHEMATICAL  INSTRUCTION 

good  and  an  end  in  itself;  he  may  or  may  not  think 
that  the  arts  and  agencies  for  the  dissemination  of 
knowledge,  as  distinguished  from  the  discovery  and 
practical  applications  of  truth,  are  important;  he  may 
or  may  not  know  that  the  art  and  the  gifts  of  the  great 
expositor  are  as  important  and  as  rare  as  those  of  the 
great  investigator  and  less  often  owe  their  success  to  the 
favor  of  accident  or  chance.  He  may  not  even  have 
seriously  considered  these  things.  He  does  know  his 
own  predilection;  and  so  strong  is  his  inclination  towards 
research  that  for  him  to  engage  in  exposition,  especially 
in  popular  exposition,  in  avocational  instruction  for 
laymen,  would  be  to  sin  against  the  authority  of  his 
vocation.  This  man,  if  he  have  intellectual  powers 
fairly  corresponding  to  the  seeming  authority  and  ur- 
gence  of  his  inner  call,  belongs  to  a  class  whose  rights 
are  peculiarly  sacred  and  whose  freedom  must  be  guarded 
in  the  interest  of  all  mankind.  It  is  not  contended 
that  every  representative  of  a  given  subject  is  under 
obligation  to  expound  it  for  the  avocational  interest 
and  enlightenment  of  laymen.  The  contention  is  that 
such  exposition  is  so  -important  a  service  that  any  uni- 
versity department  should  contain  at  least  one  man  who 
is  at  once  willing  and  qualified  to  render  it. 

I  come  now  to  the  keeping  of  my  promise.  It  is  to 
be  shown  that  the  service  is  practicable  in  the  subject 
of  mathematics  and  how  it  is  so.  Let  us  get  clearly 
in  mind  the  kind  of  persons  for  whom  the  instruction 
is  to  be  primarily  designed.  They  are  to  be  students 
of  "maturity  and  power";  they  do  not  intend  to  become 
teachers,  much  less  producers,  of  mathematics;  they 
are  probably  specializing  in  other  fields;  they  do  not 
aim  at  becoming  mathematicians;  their  interest  in 
mathematics  is  not  vocational,  it  is  avocational;  it  is 


GRADUATE   MATHEMATICAL  INSTRUCTION  191 

the  interest  of  those  whose  curiosity  transcends  the 
limits  of  any  specific  profession  or  any  specific  form  or 
field  of  activity;  each  of  them  knows  that,  whatever 
his  own  field  may  be,  it  is  penetrated,  overarched,  com- 
passed about  by  an  infinitely  vaster  world  of  human 
interests  and  human  achievements;  they  feel  its  im- 
mense presence,  the  poignant  challenge  of  it  all;  as 
specialists  they  will  win  mastery  over  a  little  part,  but 
they  have  heard  the  call  to  intelligence  and  are  seeking 
orientation  in  the  whole;  this  they  know  is  a  thing  of 
mind;  they  are  aware  that  the  essential  environment 
of  a  scholar's  life  is  a  spiritual  environment  —  the  in- 
visible and  intangible  world  of  ideas,  doctrines,  institu- 
tions, sciences  and  arts;  they  know  or  they  suspect 
that  one  of  the  great  components  of  that  world  is  mathe- 
matics; and  so,  not  as  candidates  for  a  profession  or  a 
degree,  but  in  their  higher  capacity  as  men  and  women, 
they  desire  to  learn  something  of  this  science  viewed  as 
a  human  enterprise,  as  a  body  of  human  achievements; 
and  they  are  willing  to  pay  the  price;  they  are  not  seek- 
ing entertainment,  they  are  prepared  to  work  —  to 
listen,  to  read  and  to  think. 

And  now  we  must  ask:  What  measure  of  mathe- 
matical training  is  to  be  required  of  them  as  a  prepara- 
tion? In  view  of  what  has  just  been  said  it  is  evident 
that  such  training  is  not  to  be  the  whole  of  their  equip- 
ment nor  even  the  principal  part  of  it,  but  it  is  an 
indispensable  part.  And  the  question  is:  How  much 
mathematical  knowledge  and  mathematical  discipline 
is  to  be  demanded?  I  have  no  desire  to  minimize  my 
present  task.  I,  therefore,  propose  that  only  so  much 
mathematical  preparation  shall  be  demanded  as  can 
be  gained  in  a  year  of  collegiate  study.  Most  of  them 
will,  of  course,  have  had  more;  but  I  propose  as  a  hy- 


IQ2  GRADUATE   MATHEMATICAL   INSTRUCTION 

pothesis  that  the  amount  named  be  regarded  as  an 
adequate  minimum.  But  it  does  not  include  the  differ- 
ential and  integral  calculus.  And  is  it  not  preposterous 
to  talk  of  offering  graduate  instruction  in  mathematics 
to  students  who  have  not  had  a  first  course  in  the 
calculus?  I  am  far  from  thinking  so.  A  little  reflec- 
tion will  suffice  to  show  that  in  the  case  of  such  stu- 
dents as  I  have  described  it  is  very  far  from  preposterous. 
In  my  opinion  the  absurdity  would  rather  lie  in  demand- 
ing the  calculus  of  them.  No  one  is  so  foolish  as  to 
contend  that  a  first  course  in  the  calculus  is  a  sufficient 
preparation  for  undertaking  the  pursuit  of  graduate 
mathematical  study.  But  to  suppose  it  necessary  is 
just  as  foolish  as  to  suppose  it  sufficient.  There  was 
a  time  when  it  was  necessary,  and  the  belief  that  it  is 
necessary  now  owes  its  persistence  and  currency  to  the 
inertia  then  acquired.  Formerly  it  was  necessary, 
because  formerly  all  advanced  courses,  at  least  all 
initial  courses  of  the  kind,  were  either  prolongations  of 
the  calculus,  like  differential  equations,  for  example, 
or  else  courses  in  which  the  calculus  played  an  essential 
instrumental  role  as  in  rational  mechanics,  or  the  usual 
introductions  to  function  theory  or  to  higher  geometry 
or  algebra.  But,  as  every  mathematician  knows,  that 
time  has  passed.  It  is  true  that  courses  for  which  a 
preliminary  training  in  the  calculus  is  essential  still 
constitute  and  will  continue  to  constitute  the  major 
part  of  the  graduate  offer  of  any  department  of  mathe- 
matics. And  quite  apart  from  that  consideration,  it 
seems  wise,  in  the  case  of  intending  graduate  students 
who  purpose  to  specialize  in  mathematics,  to  enforce 
the  usual  calculus  requirement  as  affording  some  slight 
protection  against  immaturity  and  the  lack  of  serious- 
ness. But  every  mathematician  knows  that  it  is  now 


GRADUATE   MATHEMATICAL   INSTRUCTION  193 

practicable  to  provide  a  large  and  diversified  body  of 
genuinely  graduate  mathematical  instruction  for  which 
the  calculus  is  strictly  not  prerequisite. 

Fortunately  it  is  just  the  material  that  is  thus  avail- 
able which  is  in  itself  best  suited  for  the  avocational 
instruction  we  are  contemplating.  As  the  calculus  is 
not  to  be  presupposed  it  goes  without  saying  that  this 
subject  must  find  a  place  in  the  scheme.  For  evidently 
an  advanced  mathematical  course  devised  and  con- 
ducted in  the  interest  of  general  intelligence  can  not 
be  silent  respecting  "the  most  powerful  weapon  of 
thought  yet  devised  by  the  wit  of  man."  Technique 
is  not  sought  and  can  not  be  given.  The  subject  is 
not  to  be  presented  as  to  undergraduates.  For  the  most 
part  these  gain  facility  with  but  little  comprehension.  It 
is  to  be  presented  to  mature  and  capable  students  who 
seek,  not  facility,  but  understanding.  Their  desire  is  to 
acquire  a  general  conception  of  the  nature  of  the  cal- 
culus and  of  its  place  in  science  and  the  history  of 
thought  —  such  a  conception  as  will  at  least  enable  them 
as  educated  men  to  mention  the  subject  without  a 
feeling  of  sham  or  to  hear  it  mentioned  without  a  feeling 
of  shame.  A  few  well-considered  lectures  should  suffice. 
At  all  events  it  would  not  require  many  to  show  the 
historical  background  of  the  calculus,  to  explain  the 
nascence  and  nature  of  the  scientific  exigencies  that 
gave  it  birth,  to  make  clear  the  concepts  of  derivative 
and  integral  as  the  two  central  notions  of  its  two  great 
branches,  and  to  present  a  few  simple  applications  of 
these  notions  to  intelligible  problems  of  typical  signifi- 
cance. Even  the  idea  of  a  differential  equation  could 
be  quickly  reached,  the  nature  of  a  solution  explained, 
and  simple  examples  given  of  physical  and  geometric 
interpretations.  As  to  the  range  and  power  of  the 


1 94  GRADUATE   MATHEMATICAL  INSTRUCTION 

calculus,  a  sense  and  insight  can  be  given,  in  some 
measure  of  course  by  a  reference  to  its  literature,  but 
much  more  effectively  by  a  few  problems  carefully 
selected  from  various  fields  of  science  and  skillfully 
explained  with  a  view  to  showing  wherein  the  methods 
of  the  calculus  are  demanded  and  how  they  serve.  Is 
not  all  this  elementary  and  undergraduate?  In  point 
of  nomenclature,  yes.  It  is  not  necessary,  however,  to 
let  words  deceive  us.  We  teach  whole  numbers  to 
young  children,  but  even  Weierstrass  was  not  aware  of 
the  logico-mathematical  deeps  that  underlie  cardinal 
arithmetic. 

The  calculus,  however,  is  hardly  the  topic  with  which 
the  course  would  naturally  begin.  A  principal  aim  of 
the  course  should  be  to  show  what  mathematics,  in  its 
inner  nature,  is  —  to  lay  bare  its  distinctive  character. 
Its  distinctive  character,  its  structural  nature,  is  that 
of  a  "  hypothetico-deductive "  system.  Probably,  there- 
fore, it  would  be  well  to  begin  with  an  exposition  of  the 
nature  and  function  of  postulate  systems  and  of  the 
great  role  such  systems  have  always  played  in  the  sci- 
ence, especially  in  the  illustrious  period  of  Greek  mathe- 
matics and  even  more  consciously  and  elaborately  in 
our  own  time.  It  is  plain  that  such  an  exposition  can 
be  made  to  yield  fundamental  insight  into  many  matters 
of  interest  and  importance  not  only  in  mathematics, 
but  in  logic,  in  psychology,  in  philosophy,  and  in  the 
methodology  of  natural  science  and  general  thought. 
The  material  is  almost  superabundant,  so  numerous 
are  the  postulate  systems  that  have  been  devised  as 
foundations  for  many  different  branches  of  geometry, 
algebra,  analysis,  Mengenlehre  and  logic.  A  general 
survey  of  these,  were  it  desirable  to  pass  them  all  in 
review,  would  not  be  sufficient.  It  will  be  necessary 


GRADUATE   MATHEMATICAL  INSTRUCTION  195 

to  select  a  few  systems  of  typical  importance  for  minute 
examination  with  reference  to  such  capital  points  as 
convenience,  simplicity,  adequacy,  independence,  com- 
patibility and  categoricalness.  The  necessity  and  pres- 
ence of  undefined  terms  in  any  and  all  systems  will 
afford  a  suitable  opportunity  to  deal  with  the  highly 
important,  much  neglected  and  little  understood  subject 
of  definition,  its  nature,  varieties  and  function,  in  light 
of  the  recent  literature,  especially  the  suggestive  han- 
dling of  the  matter  by  Enriques  in  his  "Problems  of 
Science."  A  given  system  once  thus  examined,  the 
easy  deduction  of  a  few  theorems  will  suffice  to  show 
the  possibility  and  the  process  of  erecting  upon  it  a 
perfectly  determinate  and  often  imposing  superstructure. 
And  so  will  arise  clearly  the  just  conception  of  a  mathe- 
matical doctrine  as  a  body  of  thought  composed  of  a 
few  undefined  together  with  many  defined  ideas  and  a 
few  primitive  or  postulated  propositions  with  many 
demonstrated  ones,  all  concatenated  and  welded  into  a 
form  independent  of  will  and  temporal  vicissitudes. 
Revelation  of  the  charm  of  the  science  will  have  been 
begun.  A  new  revelation  will  result  when  next  the 
possibility  is  shown  of  so  interchanging  undefined  with 
defined  ideas  and  postulates  with  demonstrated  proposi- 
tions that,  despite  such  interchange  of  basal  with  super- 
structural  elements,  the  doctrine  as  an  autonomous 
whole  will  remain  absolutely  unchanged.  But  this  is 
not  all  nor  nearly  all.  It  is  only  the  beginning  of  what 
may  be  made  a  veritable  apocalypse.  Of  great  interest 
to  any  intellectual  man  or  woman,  of  very  great  interest 
to  students  of  logic,  psychology,  or  philosophy,  should 
be  the  light  which  it  will  be  possible  in  this  connection 
to  throw  upon  the  economic  role  of  logic  and  upon  the 
constitution  of  mind  or  the  world  of  thought.  I  refer 


196  GRADUATE   MATHEMATICAL   INSTRUCTION 

especially  to  the  recently  discovered  fact  that  in  inter- 
preting a  system  of  postulates  we  are  not  restricted  to 
a  single  possibility,  but  that,  on  the  contrary,  such  a 
system  admits  in  general  of  a  literally  endless  variety 
of  interpretations;  which  means,  for  such  is  the  make- 
up of  our  Gedankenwelt,  that  an  infinitude  of  doctrines, 
widely  different  in  respect  of  their  psychological  char- 
acter and  interest,  have  nevertheless  a  common  form, 
being  isomorphic,  as  we  say,  logically  one,  though 
spiritually  many,  reposing  on  a  single  base.  And  how 
foolish  the  instructor  would  be  not  to  avail  himself  of 
the  opportunity  of  showing,  too,  in  the  same  connec- 
tion, how  various  mathematical  doctrines  that  differ 
not  only  psychologically,  but  logically  also,  are  yet 
such  that,  by  virtue  of  a  partial  agreement  in  their 
bases,  they  intersect  one  another,  owning  part  of  their 
content  jointly,  whilst  being,  in  respect  of  the  rest, 
mutually  exclusive  and  incompatible.  If,  for  example, 
it  be  some  Euclidean  system  that  he  has  been  expound- 
ing, he  will  be  able  readily  to  show  upon  how  seemingly 
slight  changes  of  base  there  arise  now  this  or  that 
variety  of  non-Euclidean  geometry,  now  a  projective  or 
an  inversion  geometry  or  some  species  or  form  of  higher 
dimensionality.  I  need  not  say  that  analogous  phe- 
nomena will  in  like  manner  present  themselves  in  other 
mathematical  fields.  And  it  is  of  course  obvious  that 
as  various  doctrines  are  thus  made  to  pass  along  in 
deliberate  panorama  it  will  be  feasible  to  point  out  some 
of  their  salient  and  distinctive  features,  to  indicate 
their  historic  settings,  and  to  cite  the  more  accessible 
portions  of  their  respective  literatures.  Naturally  in 
this  connection  and  in  the  atmosphere  of  such  a  course 
the  question  will  arise  as  to  why  it  is  that,  or  wherein, 
the  hypothetico-deductive  method  fails  of  universal 


GRADUATE   MATHEMATICAL   INSTRUCTION  1 97 

applicability.  So  there  will  be  opportunity  to  teach  the 
great  lesson  that  this  method  is  not  rudimentary7,  but 
is  an  ideal,  the  ideal  of  intellect  and  science;  to  teach 
that  mathematics  is  but  the  name  of  its  occasional 
realization;  and  that,  though  the  ideal  is,  relatively 
speaking,  but  seldom  attained,  yet  its  lure  is  universal, 
manifesting  itself  in  the  most  widely  differing  domains, 
in  the  physical  and  mechanical  assumptions  of  Newton, 
in  the  ethical  postulates  of  Spinoza,  in  our  federal  con- 
stitution, even  in  the  ten  commandments,  in  every  field 
where  men  have  sought  a  body  of  principles  to  serve 
them  as  a  basis  of  doctrine,  conduct  or  achievement. 
And  if  it  shall  thus  appear  that  mathematics  is  very 
high-placed  as  being,  in  respect  of  its  method  and  its 
form,  the  ideal  and  the  lure  of  thought  in  general,  the 
fault  must  be  imputed,  not  to  the  instructor,  but  to  the 
nature  of  things. 

In  all  this  study  of  the  postulational  method  the 
impression  will  be  gained  that  the  science  of  mathe- 
matics consists  of  a  large  and  increasing  number  of 
more  or  less  independent,  somewhat  closely  related 
and  often  interpenetrating  branches,  constituting,  not  a 
jungle,  but  rather  an  immense,  diversified,  beautifully 
ordered  forest;  and  that  impression  is  just.  At  the 
same  time  another  impression  will  be  gained,  namely, 
that  the  various  branches  rest,  each  of  them,  upon  a 
foundation  of  its  own.  This  impression  will  have  to  be 
corrected.  It  will  have  to  be  shown  that  the  branch- 
foundations  are  not  really  fundamental  in  the  science 
but  are  literally  and  genuinely  component  parts  of  the 
superstructure.  It  will  have  to  be  shown  that  mathe- 
matics as  a  whole,  as  a  single  unitary  body  of  doctrine, 
rests  .upon  a  basis  of  primitive  ideas  and  primitive 
propositions  that  lie  far  below  the  so-called  branch- 


198  GRADUATE   MATHEMATICAL  INSTRUCTION 

foundations  and,  in  supporting  the  whole,  support 
these  as  parts.  The  course  will,  therefore,  turn  to  the 
task  of  acquainting  its  students  with  those  strictly 
fundamental  researches  which  we  associate  with  such 
names  as  C.  S.  Peirce,  Schroeder,  Peano,  Frege,  Russell, 
Whitehead  and  others,  and  which  have  resulted  in 
building  underneath  the  traditional  science  a  logico- 
mathematical  sub-structure  that  is,  philosophically, 
the  most  important  of  modern  mathematical  develop- 
ments. 

It  must  not  be  supposed,  however,  that  the  instruc- 
tion must  needs  be,  nor  that  it  should  preferably  be, 
confined  to  questions  of  postulate  and  foundation, 
and  I  will  devote  the  remainder  of  the  time  at  my 
disposal  to  indicating  briefly  how,  as  it  seems  to  me, 
a  large  or  even  a  major  part  of  the  course  may 
concern  itself  with  matters  more  traditional  and  more 
concrete. 

Any  one  can  see  that  there  is  an  abundance  of  avail- 
able material.  There  is,  for  example,  the  history  and 
significance  of  the  great  concept  of  function,  a  concept 
which  mathematics  has  but  slowly  extracted  and  grad- 
ually refined  from  out  the  common  content  and  experi- 
ence of  all  minds  and  which  on  that  account  can  be 
not  only  defined  precisely  and  intelligibly  to  such  lay- 
men as  are  here  concerned,  but  can  also  be  clarified  in 
many  of  its  forms  by  means  of  manifold  examples  drawn 
from  elementary  mathematics,  from  the  elements  of 
other  sciences,  and  from  the  most  familiar  phenomena 
of  the  work-a-day  world. 

Another  available  topic  is  the  nature  and  role  of  the 
sovereign  notion  of  limit.  This,  too,  as  every  mathe- 
matician knows,  admits  of  countless  illustration  and 
application  within  the  radius  of  mathematical  knowl- 


GRADUATE   MATHEMATICAL   INSTRUCTION  1 99 

edge  here  presupposed.  In  this  connection  the  structure 
and  importance  of  what  Sylvester  called  "the  Grand 
Continuum,"  which  so  many  scientific  and  other  folk 
talk  about  unintelligently,  will  offer  itself  for  explanation. 
And  if  the  class  fortunately  contain  students  of  phil- 
osophic mind,  they  will  be  edified  and  a  little  aston- 
ished perhaps  when  they  are  led  to  see  that  the  method 
and  the  concept  of  limits  are  but  mathematicized  forms 
of  a  process  and  notion  familiar  in  all  domains  of 
spiritual  activity  and  known  as  idealization.  Not 
improbably  some  of  the  students  will  be  sufficiently 
enterprising  to  trace  the  mentioned  similitude  in 
some  of  its  manifestations  in  natural  science,  in  psy- 
chology, in  philosophy,  in  jurisprudence,  in  literature 
and  in  art. 

I  have  not  mentioned  the  modern  doctrine  variously 
known  as  Mengenlehre,  Mannigfaltigkeitslekre,  the  theory 
of  point-sets,  assemblages,  manifolds,  or  aggregates: 
a  live  and  growing  doctrine  in  which  expert  and  layman 
are  about  equally  interested  and  which,  like  a  subtle  and 
illuminating  ether,  is  more  and  more  pervading  mathe- 
matics in  all  its  branches.  For  the  avocational  in- 
struction of  lay  students  of  "maturity  and  power"  how 
rich  a  body  of  material  is  here,  with  all  its  fascinating 
distinctions  of  discrete  and  continuous,  finite  and  in- 
finite, denumerable  and  non-denumerable,  orderless, 
ordered,  and  well-ordered,  and  with  its  teeming  host 
of  near-lying  propositions,  so  interesting,  so  illuminating, 
often  so  amazing. 

Finally,  but  far  from  exhausting  the  list,  it  remains 
to  mention  the  great  subjects  of  invariants  and  groups. 
Both  of  them  admit  of  definition  perfectly  intelligible  to 
disciplined  laymen;  both  admit  of  endless  elementary 
illustration,  of  having  their  mutual  relations  simply 


200  GRADUATE    MATHEMATICAL   INSTRUCTION 

exemplified,  of  being  shown  in  historic  perspective, 
and  of  being  strikingly  connected,  especially  the  notion 
of  invariance,  with  the  dominant  enterprise  of  man: 
his  ceaseless  quest  for  the  changeless  amid  the  turmoil 
and  transformation  of  the  cosmic  flux. 


THE  SOURCE  AND  FUNCTIONS  OF  A 
UNIVERSITY1 

IN  returning  hither  from  near  and  far  to  join  in  cele- 
brating the  seventy-fifth  anniversary  of  the  founding  of 
their  academic  birthplace  and  home,  the  alumni,  the 
sons  and  daughters  of  this  institution,  have  not  come  to 
congratulate  an  eld-worn  mother  upon  the  continuance 
of  her  years  beyond  the  Psalmist's  allotment  of  three 
score  and  ten  nor  to  comfort  her  in  the  sorrows  of  age. 
Their  assembling  is  due  to  other  sentiments  and  owns 
another  mood.  They  have  come  as  beneficiaries  in 
order  to  pay,  for  themselves  and  for  the  many  absent 
ones  whom  they  have  the  honor  to  represent,  a  tribute 
of  gratitude,  loyalty  and  love  to  a  noble  benefactress 
who,  notwithstanding  her  wisdom  and  fame,  yet  is 
literally  in  the  early  morning  of  her  life.  For  it  is  not 
written,  nor  ordained  in  the  scheme  of  things,  that,  in 
respect  of  years,  the  life  of  a  university  shall  be  as  a 
tale  that  is  told  or  a  watch  in  the  night.  It  is  indeed  a 
living  demonstration  of  the  greatness  of  man,  bearing 
witness  to  his  superiority  even  over  death,  that  men  and 
women,  though  they  themselves  must  die,  yet  may, 
whilst  they  live,  create  ideals  and  institutions  that 
survive.  A  college  or  a  university  may  indeed  have 
been  as  a  benignant  mother  to  a  thousand  academic 

1  An  address  delivered  June  3,  1014,  at  the  celebration  of  the  seventy- 
fifth  anniversary  of  the  founding  of  the  University  of  Missouri.  Printed 
in  The  Columbia  University  Quarterly,  March, 


202          SOURCE   AND   FUNCTIONS   OF   A   UNIVERSITY 

generations  and  yet  be  younger  than  her  youngest 
child.  Unlike  man  the  individual,  a  university  is,  like 
man  the  race,  immortal.  The  age  of  three  score  and 
fifteen  in  the  life  of  an  immortal  institution  is  a  mere 
beginning.  In  emphasizing  this  consideration  it  is  not 
my  intention  to  suggest  or  imply  that  the  services  ren- 
dered by  the  University  of  Missouri  have  necessarily 
been,  because  of  her  youth,  meagre  or  ineffectual  or 
immature.  On  the  contrary  I  maintain  that  her  serv- 
ices to  the  people  of  this  state  have  been  beyond  com- 
putation and  that  already  her  spiritual  achievements 
constitute  the  chief  glory  of  a  great  commonwealth. 
Is  it  the  alumni  only  who  owe  her  grateful  allegiance? 
Is  the  beneficence  of  an  institution  of  learning  exclu- 
sively or  even  mainly  confined  to  the  relatively  few  who 
dwell  for  a  season  in  her  immediate  presence,  who  touch 
the  hem  of  her  garment,  come  into  personal  contact 
with  her  scholars  and  teachers  and  receive  her  degrees? 
Far  from  it.  Far  from  being  the  sole  or  the  principal 
beneficiaries  of  a  university,  the  alumni  are  simply 
among  the  more  potent  instrumentalities  for  extending 
her  ministrations  to  ever  wider  and  wider  circles.  The 
sun,  we  say,  is  far  off  yonder  in  the  heavens.  But 
strictly  speaking  the  sun  really  is  wherever  he  shines. 
Where  is  the  University  of  Missouri?  At  Columbia,  we 
say,  and  the  speech  is  convenient.  But  it  is  juster  to 
say  that,  owing  to  the  pervasiveness  of  her  light  and 
inspiration,  the  University  of  Missouri  in  a  measure 
now  is,  and  in  larger  and  larger  measure  will  come  to 
be,  in  every  home  and  school,  in  every  factory  and  field, 
in  every  mine  and  shop,  in  every  council  chamber,  in 
every  office  of  charity,  or  medicine,  or  law,  in  all  the 
places  near  or  remote  where  within  the  borders  of  this 
beautiful  state  children  play  and  men  and  women 


SOURCE   AND   FUNCTIONS  OF  A   UNIVERSITY          203 

think  and  love,  suffer  and  hope,  aspire  and  toil.  Nay, 
by  the  researches  and  publications  of  her  scholars  and 
by  the  migrations  of  those  she  has  inspired  and  dis- 
ciplined, the  University  of  Missouri  to-day  lives  and 
moves  abroad,  mingling  her  presence  with  that  of  kin- 
dred agencies,  not  only  in  every  state  of  the  union  but 
in  many  other  quarters  of  the  civilized  world. 

It  is  not  my  purpose  to  review  the  history  of  her 
aspirations  and  struggles  nor  to  relate  the  thrilling 
story  of  her  triumphs.  I  conceive  that  the  central 
motive  of  our  assembling  here  is  not  so  much  to  praise 
the  University  for  what  she  has  already  accomplished  as 
to  renew  our  devotion  to  her  high  emprize,  to  congratu- 
late her  upon  her  solid  attainments,  to  rejoice  in  her 
divine  discontent  and  spirit  of  progressiveness,  to  deepen 
and  enlarge  our  conception  of  her  mission  and  destiny, 
and  especially  to  remind  ourselves  of  the  principles, 
the  faith  and,  above  all,  the  ideals  to  which  she  owes 
her  birth,  her  continuity,  her  responsibilities,  and  her 
power. 

What  is  a  university?  How  shall  we  conceive  that 
marvelous  thing  which,  though  having  a  local  habitation 
and  a  name  and  seeming  to  dwell  in  houses  made  by 
human  hands,  yet  contrives  to  be  omnipresent;  per- 
vading the  abodes  of  men  everywhere  throughout  a 
state,  a  nation  or  a  world,  like  a  divine  ether;  subtly, 
gently,  unceasingly,  increasingly  ministering  to  their 
hearts  and  minds  healing  counsels  and  the  mysterious 
grace  of  light  and  understanding?  What  is  it?  Is  it 
something,  an  agency  or  an  influence,  that  can  be 
denned?  We  know  that  it  is  not.  We  know  that  the 
really  great  things  of  the  world,  the  things  that  live 
and  grow  and  shine,  the  things  that  give  to  life  its 
interests  and  its  worth,  one  and  all  elude  formulation. 


204          SOURCE  AND   FUNCTIONS   OF   A   UNIVERSITY 

Yet  it  is  just  these  things,  beauty  and  love,  poetry  and 
thought,  religion  and  truth  and  mind,  it  is  precisely 
these  great  indefinables  of  life  that  we  may  learn, 
through  experience  and  discipline,  to  know  best  of  all. 
And  so  it  is  with  what  we  mean  or  ought  to  mean  by 
a  university.  What  a  university  is  no  one  can  define, 
but  all  may  in  a  measure  come  to  know.  •  By  pon- 
dering its  principles,  by  contemplating  its  ideals,  by 
examining  its  aims,  activities  and  fruits,  above  all  by 
sharing  in  its  spirit  and  aspiration,  we  may  at  length 
win  a  conception  of  it  that  will  fill  our  minds  with  light 
and  our  hearts  with  devotion. 

Where  such  a  conception  reigns  a  university  will 
flourish.  But  there  is  no  conception  more  difficult  for 
a  people  to  acquire.  It  is  not  a  spontaneous  growth, 
springing  up  like  a  weed,  but  requires  careful  planting 
and  cultivation.  Such  is  the  husbandry  to  which  a 
university  must  perpetually  devote  itself  as  the  essen- 
tial precondition  to  the  prosperous  exercise  and  advance- 
ment of  all  its  other  functions,  and  the  husbandry  is 
not  easy.  Especially  in  our  American  communities 
where  universities  must  appeal  for  support  to  the  in- 
telligence of  a  democratic  people,  there  is  no  service 
more  important  or  more  difficult  to  render  than  that 
which  consists  in  teaching  us  to  know  what  a  university 
really  is  and  what  it  signifies  alike  for  developing  the 
material  resources  of  the  world  and  for  the  spiritualizing 
of  man.  And  thus  there  devolves  upon  a  university, 
especially  in  the  beginning  of  its  career,  the  necessity 
of  performing  a  kind  of  miracle:  without  adequate 
support,  either  material  or  moral,  it  must  yet  find 
strength  to  teach  us  to  give  it  both.  The  lesson  is 
one  that  takes  long  and  long  to  teach  because  it  is  one 
that  takes  long  and  long  to  learn. 


SOURCE  AND  FUNCTIONS  OP  A  UNIVERSITY          2O$ 

It  is  a  great  mistake  to  imagine  that  a  university  is 
an  essentially  modern  thing.  In  spirit,  in  idea  and 
essence,  it  is  modern  only  in  the  sense  in  which  forces 
and  ideals  that  are  eternal  are  always  modern,  as  they 
are  always  ancient.  We  should  not  forget  that  even 
the  name  University  —  so  suggestive  of  the  infinite- 
world  which  it  is  the  aim  of  these  institutions  to  sub- 
jugate to  the  understanding  and  uses  of  man  —  even 
the  name,  in  its  modern  scholastic  sense,  has  had  a 
history  of  more  than  a  thousand  years.  But  we  know 
that  the  institution  itself,  the  thing  that  bears  the  name, 
owns  an  antiquity  far  more  remote.  A  few  years  ago, 
standing  upon  the  Acropolis  of  Athens,  gazing  pensively 
about  upon  the  hallowed  scene  where  culminated  the 
genius  of  the  ancient  world,  a  friend,  pointing  towards 
the  spot  near  by  where  for  fifty  years  Plato  taught  in 
the  grove  of  Academe,  said  to  me,  yonder,  yonder 
is  the  holy  ground  where  was  made  the  first  attempt 
to  organize  higher  education  in  the  western  world.  The 
remark,  which  was  just  enough,  was  indeed  impressive.  It 
is  easy,  however,  to  misunderstand  its  significance  and 
to  exaggerate  its  importance.  So  many  of  the  most 
precious  elements  of  our  civilization  trace  their  lineage 
back  to  the  creative  activity  of  ancient  Greece  that  we 
are  naturally  tempted  to  imagine  we  may  find  there 
also  the  source  and  origin  of  those  aims,  activities  and 
ideals  which  constitute  what  we  today  call  a  univer- 
sity. Such  imagining,  however,  is  vain.  The  originals, 
the  first  organizations,  we  may  possibly  find  there  at  a 
definite  time  and  place,  but  not  the  origin,  not  the 
source,  not  the  nascence  of  the  birth-giving  and  life- 
sustaining  power.  For  this  must  account,  not  only  for  the 
universities  of  our  time,  but  for  the  great  school  of  Plato 
as  well.  What,  then,  and  where  is  the  secret  spring? 


206          SOURCE   AND   FUNCTIONS   OF  A   UNIVERSITY 

Shall  we  seek  it  in  a  sense  of  need?  Necessity  is 
indeed  a  keen  spur  to  invention  and  is  the  mother  of 
many  things.  But  necessity  is  not  the  mother  of  uni- 
versities. The  beasts  flourish  and  propagate  their  kind 
without  the  help  of  institutions  of  learning,  and  without 
such  help  a  similar  existence  is  possible  to  men.  Uni- 
versities are  not  essential  to  life  nor  to  animal  pros- 
perity. They  are  not  creatures,  they  are  creators, 
of  need.  We  do  indeed  nowadays  hear  much  of  the 
services  they  render,  and  it  is  right  that  we  should,  for 
they  minister  constantly  and  everywhere  to  countless 
forms  of  need.  But  the  needs  they  supply  are  in  the 
main  needs  that  they  have  first  produced,  multiplied 
desires  and  aspirations,  new  propensions  of  mind  awak- 
ened to  new  life,  lifted  by  education  to  higher  levels 
and  ampler  possibilities  of  being.  No,  the  origin,  the 
source  we  are  seeking,  the  principle  of  explanation,  is 
no  human  contrivance  nor  institution  nor  sense  of  need. 
It  is  that  sovereign  urgency,  at  once  so  strange  and  so 
familiar,  that  drives  us  to  seek  it;  it  is  the  lure  of 
wisdom  and  understanding,  of  beauty  and  light;  a 
certain  divine  energy  in  the  world,  at  once  a  cosmic 
force  and  a  human  faculty,  constituting  man  divine 
in  constituting  him  a  seeker  of  truth  and  a  lover  of 
harmony  and  illumination. 

Has  it  an  epoch  and  a  name?  It  has  both.  In 
accordance  with  the  modern  doctrine  of  evolution  the 
greatest  events  upon  our  planet  occurred  long  before 
the  beginnings  of  recorded  history.  For  according  to 
that  doctrine  there  must  have  come  a  time,  long,  long 
ago,  when  in  what  was  a  world  of  matter  there  began 
to  be  mind,  in  what  was  a  world  of  motion  there  began 
to  be  emotion,  and  the  blind  dominion  of  force  was 
invaded  by  personality.  Among  all  those  marvels  of 


SOURCE  AND   FUNCTIONS   OP   A   UNIVERSITY          207 

prehistoric  history,  the  supreme  event  was  that  one  but 
for  which  this  world  had  been  a  world  devoid  of  mystery 
and  devoid  of  truth  —  I  mean  the  advent  of  Wonder. 
With  the  advent  of  wonder  came  the  sense  of  mystery, 
the  lure  of  truth,  the  sheen  of  ideality,  the  dream  of  the 
perfect  and,  with  these,  the  potence  and  promise  of 
research  and  creativeness  with  all  their  endless  progeny 
of  knowledge  and  wisdom  and  science  and  art  and 
philosophy  and  religion.  These  things,  children  of  the 
spirit,  offspring  of  wonder,  these  things  are  the  interests 
which  it  is  the  divine  prerogative  of  universities  to  serve, 
and  the  universities  ultimately  derive  their  own  exist- 
ence, their  sustenance  and  their  power  from  the  same 
mother  that  gives  their  charges  birth.  A  genuine  uni- 
versity is  thus  the  offspring  and  the  appointed  agent  of 
the  spirit  of  inquiry;  it  is  the  offspring,  expression  and 
servant  of  that  imperious  curiosity  which  in  a  measure 
impels  all  men  and  women,  but  with  an  urgency  like 
destiny  literally  drives  men  and  women  of  genius,  to 
seek  to  know  and  to  teach  to  their  fellows  whatsoever 
is  worthy  in  all  that  has  been  discovered  or  thought, 
spoken  or  done  in  the  world,  and  at  the  same  time 
seeks  to  extend  the  empire  of  understanding  endlessly 
in  all  directions  throughout  the  infinite  domain  of  the 
yet  uncharted  and  unknown.  That  high  commission 
is  at  once  a  university's  charter  of  freedom  and  the 
definition  of  her  functions  and  her  obligations.  These 
are,  on  the  one  hand,  to  teach  —  to  teach  with 
no  restrictions  save  those  prescribed  by  decency  and 
candor  —  and,  on  the  other  hand,  to  foster  and 
prosecute  research  —  research  in  any  and  all  subjects 
or  fields  to  which  the  leading  or  the  stress  of 
curiosity  may  draw  or  impel.  In  so  far  as  the 
great  commonwealth  of  Missouri  makes  ample  pro- 


208          SOURCE   AND   FUNCTIONS   OF  A   UNIVERSITY 

vision  for  the  exercise  of  these  functions  and  for  the 
discharge  of  these  obligations,  to  that  extent  she 
may  be  said  to  cooperate  with  the  divine  energy 
of  the  world  in  the  maintenance  of  a  genuine  uni- 
versity. 


RESEARCH  IN  AMERICAN  UNIVERSITIES1 

THE  present  writer  has  been  asked  to  deal  briefly 
with  the  question  of  research  in  American  universities. 
The  subject  is  an  immense  one,  and  the  following  dis- 
cussion makes  no  pretense  of  being  exhaustive.  It 
aims  merely  to  present  the  problem  again,  to  emphasize 
again  its  importance,  and  to  point  out  once  more  some 
of  its  harder  conditions  and  some  of  the  principles  and 
distinctions  involved  in  any  serious  attempt  at  its 
solution. 

The  problem  may  not  be  easy  to  appreciate,  but  it 
is  at  all  events  easy  to  state.  It  is  the  problem  of 
securing  in  our  universities  suitable  provision  for  the 
work  of  research  or  investigation  and  productivity.  For 
a  generation  the  great  majority  of  the  ablest  men  in 
our  universities  have  regarded  that  problem  as  the 
most  urgent  and  important  educational  problem  con- 
fronting these  institutions  and  the  American  people. 
Meanwhile,  something  has  been  done  towards  a  solu- 
tion. But  none  of  the  universities  has  secured  ade- 
quate provision,  and  the  majority  of  them  but  little  or 
none  at  all.  In  the  abstract,  the  problem  is  simple  and 
the  solution  is  easy:  given  a  body  of  able  and  enthu- 
siastic men,  provide  them  with  proper  facilities,  afford 
them  opportunity  to  devote  their  powers  continuously 
to  the  prosecution  of  research,  and  the  thing  is  done. 
But  in  the  concrete  it  is  exceedingly  difficult,  being 

1  Printed  in  The  Bookman,  May,  1906. 


210  RESEARCH   IN  AMERICAN  UNIVERSITIES 

frightfully  complicated  with  our  whole  institutional 
history  and  life,  in  particular  with  our  educational 
traditions  and  tendencies,  with  the  prevailing  plan  of 
university  organisation,  and  especially  with  the  char- 
acteristic temper,  ideals  and  ambitions  of  the  American 
people. 

Somebody  besides  our  foreign  friends  and  critics 
ought  to  tell  the  truth  about  American  education  and 
American  universities.  Our  people  have  never  ceased 
to  believe  in  education.  Our  belief  has  not  always  been 
intelligent.  We  have  been  prone  to  ascribe  to  educa- 
tion efficacies  and  potencies  that  do  not  belong  to  any 
human  agency  or  institution.  But  our  faith  in  it, 
though  not  always  critical  or  enlightened,  has  been  deep, 
implicit  and  abiding;  and  we  have  diligently  pursued 
it,  generally  as  a  means  no  doubt,  but  sometimes  as 
an  end,  and  occasionally  as  a  thing  in  itself  more  pre- 
cious than  power  and  gold.  In  all  this  we  have  been, 
quite  unconsciously  and  contrary  to  all  appearances, 
very  humble.  We  have  been  content  to  educate  our- 
selves with  knowledge  discovered  by  others  and  to 
nourish  ourselves  with  doctrines  and  truths  produced 
only  by  the  spiritual  activity  of  other  lands.  We  may 
have  been  vain  but  we  have  not  been  proud.  Besides 
a  marvelous  practical  sense  we  have  had,  in  degree 
quite  unsurpassed,  two  of  the  elements  of  genius,  — 
intellectual  energy  and  intellectual  audacity;  and  by 
means  of  these  we  have  created  a  material  civilisation 
so  obtrusive,  so  elaborate  and  so  efficient  as  to  amaze 
the  world.  But  now  at  length  there  begin  to  appear 
the  indicia  of  change,  of  change  for  the  better.  A  new 
day  has  dawned.  The  sun  is  not  yet  risen  high,  but 
it  is  rising.  We  have  begun  to  suspect  that  genuine 
civilisation  is  essentially  an  a.ffair  of  the  spirit,  that  it 


RESEARCH   IN   AMERICAN   UNIVERSITIES  211 

can  not  be  borrowed  nor  imported  nor  improvised  nor 
appropriated  from  without,  but  that  it  is  a  growth  from 
within,  an  efflorescence  of  mind  and  soul,  and  that  its 
highest  tokens  are  not  soldiers  but  savants,  not  the 
purchasers  and  admirers  of  art  but  artists,  not  mere 
retailers  of  knowledge  nor  teachers  of  the  familiar  and 
the  known,  but  discoverers  of  the  unknown,  not  mere 
inventors  but  men  of  science.  And  so  we  have  begun 
to  feel  our  way  towards  the  establishment  of  true 
universities,  that  is  to  say  of  institutional  centres  for 
the  activity  of  the  human  spirit,  and  of  organs,  the 
most  potent  yet  invented  by  human  society,  for  giving 
effect  to  the  noblest  instinct  of  man,  "the  civilisation- 
producing  instinct  of  truth  for  truth's  sake." 

Just  here  we  encounter  a  great  danger.  For  a  gen- 
eration our  progress  in  the  matter  has  been  so  swift 
that  both  the  universities  themselves  and  the  edu- 
cated public  opinion  upon  which  in  our  democratic 
society  their  support  and  advancement  ultimately  de- 
pend, are  in  danger  of  greatly  overestimating  it,  and 
that  would  be  a  misfortune.  Absolutely  the  progress 
has  indeed  been  great,  but  relatively  and  judged  by  the 
very  highest  standards,  it  has  not.  It  is  not  first  nor 
mainly  a  question  of  achievements,  of  things  done.  It 
is  a  quesion  of  ideals,  of  standards  and  aspirations.  A 
clear  concept  of  a  great  university  unconsciously  serving 
the  highest  interests  of  man  by  absolute  devotion  to 
Truth  for  its  own  sake  and  without  extraneous  motive, 
end  or  aim,  does  not  yet  exist  in  the  mind  of  the  Amer- 
ican public  and  is  not  yet  incarnate  in  any  of  its  institu- 
tions. Our  universities  are  young,  strong  and  robust. 
They  are  full  of  potence  and  promise.  But  they  have 
not  yet  impressed  their  own  imperfect  ideals  upon  the 
people;  they  have  not  yet  given  forth  the  light  ncces- 


212  RESEARCH   IN  AMERICAN   UNIVERSITIES 

sary  for  their  own  proper  beholding  and  appreciation. 
Their  perfections  and  their  imperfections  alike,  remain 
obscure.  The  old  colleges  about  which  as  about  nuclei 
some  of  our  universities  have  been  formed  have  done 
much  to  leaven  and  temper  the  American  mind  and  to 
subdue  it  to  the  influences  of  beauty  and  truth.  Cor- 
responding services  have  not  yet  been  rendered  by  our 
universities  as  such.  No  one  can  doubt  that  they  are 
destined  to  assume  in  future  the  permanent  leadership, 
and  to  exercise  a  controlling  formative  influence,  in  all 
that  goes  to  deepen  thought  and  to  exalt  and  refine 
standards,  character,  and  taste.  At  present,  however, 
they  are  themselves  in  the  formative  and  impressionable 
stage,  resembling  improvisations  in  some  respects;  and 
to  understand  them,  to  see  clearly  both  what  they  are 
and  what  they  are  not,  it  is  necessary  to  regard  them 
as  being  at  the  present  time  less  the  producers  than 
the  products  of  our  civilisation. 

So  regarded,  they  are  seen  to  embody  and  to  reflect 
alike  the  merits  and  the  defects  of  their  progenitor. 
Like  the  latter  they  are  unsurpassed  in  boldness,  in 
energy  and  in  enthusiasm,  and  their  genius  has  been 
mainly  directed  to  material  and  outer  ends.  Their 
first  and  chief  concern  has  been  with  the  physical  and 
exterior,  with  buildings  and  grounds  and  instruments 
and  laboratories,  and  while  their  material  equipment  is 
still  far  from  adequate,  it  has  already  evoked  astonished 
and  admiring  commentary  from  visiting  scholars  of 
European  seats  of  learning.  Like  the  civilisation  whence 
they  have  sprung,  our  universities  are  intensely  modern 
and  up-to-date,  and  they  are  intensely  democratic  in 
everything  but  management;  they  set  great  store  by 
organisation,  exalt  the  function  of  administration,  and 
tend  to  be  regarded,  to  regard  themselves,  and  in  fact 


RESEARCH   IN   AMERICAN  UNIVERSITIES  213 

to  be,  as  vast  and  complicate  machines  or  industrial 
plants  naturally  demanding  the  control  of  centralised 
authority.  They  have  but  little  sentiment;  they  are 
almost  devoid  of  sacred  and  hallowing  traditions,  of 
great  and  illustrious  recollections;  there  is  in  and  about 
them  nothing  or  but  little  of  "the  shadow  and  the  hush 
of  a  haunted  past."  They  have  no  antiquity.  In  them 
the  utilitarian  spirit,  having  learned  the  lingo  of  service, 
contrives  to  receive  an  ample  share  of  honour,  and  the 
Genius  of  Industry  that  has  transformed  our  land  into 
an  abode  of  wealth  and  for  generations  assigned  an 
attainable  upper  limit  to  a  people's  aspiration,  shapes 
educational  policy,  holds  and  wields  the  balance  of 
power.  The  classic  distinctions  of  good,  better  and  best 
in  subjects  and  motives  of  study  receive  but  slight  re- 
gard. The  traditional  hierarchy  of  educational  values 
and  the  ascending  scale  of  spiritual  worths  have  fallen 
into  disrepute.  All  things  have  been  leveled  up  or 
leveled  down  to  a  common  level;  so  that  the  workshop 
and  the  laboratory,  schools  of  engineering,  of  agri- 
culture and  of  the  classics,  the  library,  the  model  dairy 
and  departments  of  architecture  and  music,  exist  side 
by  side.  In  at  least  one  institution,  so  it  is  reported, 
the  professor  of  poetry  rubs  shoulders  with  the  pro- 
fessor of  poultry.  No  wonder  that  a  distinguished 
critic  has  said  that  some  of  our  biggest  universities 
appear  as  hardly  more  than  episodes  in  the  wondrous 
maelstrom  of  our  industrial  life. 

Thus  it  appears  that  the  American  university,  child 
of  a  predominantly  material  and  industrial  civilisa- 
tion half-blindly  aspiring  to  higher  things,  strikingly 
resembles  its  parent.  Begotten  in  the  hope  that  it 
would  be  as  a  saviour  and  rescue  us  from  our  national 
idols  and  respectable  sins,  it  straightway  became  their 


214  RESEARCH   IN  AMERICAN   UNIVERSITIES 

most  enlightened  servant  and  lent  them  the  sanction 
and  the  support  of  its  honoured  name.  It  is  by  no 
means  contended  that  this  fact  is  the  whole  truth. 
Our  universities  are  not  entirely  devoted  to  the  service 
of  industry;  they  are  not  wholly  committed  to  teaching 
youth  the  known  from  utilitarian  motives  and  for  imme- 
diate and  practical  ends;  they  are  not  exclusively 
concerned  with  the  applications  of  science;  out  of  gen- 
eral devotion  to  the  Useful,  something  is  saved  for  the 
True;  science  is  not  always  regarded  as  a  commodity; 
the  judgment  of  the  great  Jacobi  is  sometimes  recog- 
nised as  just:  "The  unique  end  ot  science  is  the  honour 
of  the  human  spirit."  And  it  is  a  pleasure  to  be  able 
to  proclaim  the  fact  that  in  a  few  of  our  universities 
something  like  a  home  has  been  provided  for  the  spirit 
of  research  and  that  by  its  activity  there,  American 
genius  has  had  a  share  in  extending  the  empire  of  light, 
in  enlarging  the  domain  of  the  known,  in  astronomy,  in 
physics,  in  mathematics,  in  the  science  of  mind,  in  biol- 
ogy, in  criticism,  in  economics,  in  letters,  in  almost  all 
of  the  great  fields  where  the  instinct  of  truth  for  the 
sake  of  truth  contends  against  the  dark.  In  this  clear 
evidence  of  our  growing  freedom  and  exaltation,  let  us 
rejoice;  but  let  us  be  candid  also.  Let  us  admit  that 
we  have  only  begun  the  higher  service  of  the  soul;  let 
us  confess  in  becoming  humility  that,  in  comparison 
with  our  wealth,  our  numbers,  our  energies  and  our 
talents,  in  comparison,  too,  with  the  intellectual  achieve- 
ments of  some  other  peoples  and  other  lands,  the  service 
we  have  rendered  to  Science  and  Art  and  Truth  is 
meagre. 

Why  such  emptiness,  such  poverty,  such  meagreness 
in  the  fruits  of  the  highest  activity?  The  immediate 
cause  is  easy  to  find.  It  is  not  incompetence  nor  lack 


RESEARCH  IN  AMERICAN  UNIVERSITIES 

of  genius  in  our  university  faculties.  These  are  not 
inferior  to  the  best  in  the  world.  It  is  not  mainly  due, 
as  is  often  said,  to  inadequacy  of  material  compensa- 
tion, though  one  of  the  greatest  of  living  physicists, 
Professor  J.  J.  Thompson,  has  told  us  truly  that  Amer- 
ican men  of  science  receive  less  remuneration  than  their 
colleagues  in  any  other  part  of  the  world.  The  cause 
in  question  is  simple:  lack  of  opportunity.  The  diffi- 
culty is  near  at  hand.  It  inheres  in  the  composition 
and  organisation  of  our  universities.  Most  of  these  are 
built  about  and  upon,  and  largely  consist  of,  immense 
undergraduate  schools  thronged  by  young  men  mainly 
bent  upon  practical  aims  and  neither  qualified  nor 
intending  to  qualify  for  the  work  of  investigation.  The 
interests  of  these  schools  are  naturally  the  paramount 
concern.  The  great  and  growing  burdens  of  adminis- 
tration tend  to  distribute  themselves  among  the  pro- 
fessors. These  have,  besides,  to  give  the  most  and  the 
best  of  their  energies  to  elementary  teaching,  to  teach- 
ing, that  is,  which  does  not  pertain  to  a  university 
proper  but  to  gymnasia  and  Iyc6es  —  a  worthy,  impor- 
tant, necessary  kind  of  work,  but  a  kind  that  drains 
off  the  energy  in  non-productive  channels  and  tends  to 
form  and  harden  the  mind  of  those  engaged  in  it  about 
a  small  group  of  simpler  ideas.  What  is  left,  what  can 
be  left,  of  spirit,  of  energy,  of  opportunity,  for  the 
arduous  work  of  research?  One  man  attempting  the 
enterprise  of  three:  administration,  elementary  teach- 
ing, discovery  and  creative  work.  Who  can  suitably 
characterise  the  absurdity?  Who  can  compute  the 
wickedness  of  the  waste  in  the  impossible  attempt  to 
effect  daily  the  demanded  transition  from  mood  to  mood? 
A  mind,  by  prolonged  effort,  at  length  immersed  in 
the  depths  of  a  profound  and  difficult  investigation  - 


2l6  RESEARCH   IN  AMERICAN  UNIVERSITIES 

how  poignant  the  pain  of  interruption,  the  rending  of 
continuity,  the  rude  disturbance  of  poise  and  concentra- 
tion. How  easy  to  fail  of  due  respect  for,  because  it 
is  so  easy  not  to  understand,  the  creative  mood,  obliv- 
ious to  the  outer  world,  the  brooding  "maternity  of 
mind,"  more  delicate  than  fabric  of  gossamer,  of  infinite 
subtlety,  of  infinite  sensitiveness,  a  woven  psychic  struc- 
ture finer  than  ether  threads;  and  how  easy  to  forget 
that  a  sudden  alien  call  may  disturb  and  jar  and  even 
destroy  the  structure. 

Little  excuse,  then,  have  we  to  wonder  at  the  recent 
words  of  Professor  Bjerknes,  of  the  chair  of  mechanics 
and  mathematical  physics  in  the  University  of  Stock- 
holm, and  non-resident  lecturer  in  mathematical  physics 
in  Columbia  University,  who,  in  his  farewell  address 
to  his  American  colleagues,  assembled  to  do  him  honour, 
spoke  substantially  as  follows: 

"I  have  been  much  impressed  with  the  material  equipment  of  your  uni- 
versities, with  your  splendid  buildings,  with  the  fine  instruments  you  have 
placed  in  them,  and  with  the  enthusiasm  of  the  men  I  have  found  at  work 
there.  But  I  hope  you  will  pardon  me,  gentlemen,  for  saying,  as  I  must  say, 
that,  when  I  found  you  attempting  serious  investigation  with  the  remnants 
of  energy  left  after  your  excessive  teaching  and  administrative  work,  I 
could  not  help  thinking  you  did  not  appreciate  the  fact  that  the  finest 
instruments  in  those  buildings  are  your  brains.  I  heard  one  of  you  counsel 
his  colleagues  to  care  for  the  astronomical  instruments  lest  these  become 
strained  and  cease  to  give  true  results.  Allow  me  to  substitute  brain  for 
telescope,  and  to  exhort  you  to  care  for  your  brains.  I  have  been  aston- 
ished to  find  that  some  of  you,  in  addition  to  much  executive  work,  teach 
from  ten  to  fifteen  and  even  more  hours  per  week.  I  myself  teach  two  hours 
per  week,  and  I  can  assure  you  that,  if  I  had  been  required  to  do  so  much 
of  it  as  you  do,  you  never  would  have  invited  me  to  lecture  here  in  a  diffi- 
cult branch  of  science.  That,  gentlemen,  is  the  most  important  message  I 
can  leave  with  you." 

Such,  then,  is  the  situation.  No  need  that  we  should 
behold  it  in  picture  drawn  by  foreign  hand.  We  need 
no  copy.  The  original  lies  before  us  in  all  its  proper- 


RESEARCH  IN  AMERICAN  UNIVERSITIES  217 

tions.  The  challenge  addresses  itself  at  once  to  our  pride 
and  to  our  practical  sense.  Of  all  peoples,  we,  it  would 
seem,  should  feel  the  challenge  most  keenly,  for  the 
problem  is  a  problem  in  freedom.  It  demands  the 
emancipation  of  American  genius;  it  demands  pro- 
vision of  free  and  ample  opportunity  for  the  highest 
activity  of  our  highest  talent. 

Hope  of  solution  lies  in  division  of  labour.  Our  uni- 
versities and  the  people  they  represent  must  reduce 
their  exactions.  For  three  men's  work,  three  must  be 
provided.  There  must  be  men  to  administer  and  men 
to  teach  and  men  to  investigate.  Three  varieties  of 
service,  entirely  compatible  in  kind,  entirely  incompat- 
ible as  co-ordinate  vocations  combined  in  one.  Any 
one  of  them  may  be  as  an  avocation  to  another  of  the 
three,  but  only  so  of  choice  and  not  by  compulsion. 
No  invidious  comparisons  are  implied.  The  distinctions 
are  not  of  greater  and  less;  they  are  matters  of  economy 
in  the  domain  of  mind.  The  great  administrator  is  not 
a  clerk  nor  an  amanuensis;  he  is  a  man  of  constructive 
genius,  a  creator.  The  great  teacher  is  not  a  pedagogue; 
he  is  a  source  of  inspiration  and  of  aspiration,  produc- 
ing children  of  the  spirit  by  "the  urge  and  ardor"  of 
a  deep  and  rich  and  enlightened  personality;  he  was 
in  the  mind  of  Goethe  when  he  said  of  Winckelmann 
that  "from  him  you  learned  nothing,  but  you  became 
something."  And  the  great  investigator  is  not  a  mere 
collector  and  recorder  of  facts;  he  is  a  discoverer,  a  dis- 
closer,  of  the  harmonies  and  the  invariance  hid  beneath 
the  surface  of  seeming  disorder  and  of  ceaseless  change. 
The  three  great  powers  are  compatible,  and  are  usually 
found  united  in  a  single  gigantic  personality,  just  as  the 
ordinary  administrator  and  ordinary  teacher  and  ordi- 
nary investigator  compose  one  unit  of  mediocrity. 


2l8  RESEARCH   IN  AMERICAN  UNIVERSITIES 

It  is  perfectly  evident  that  the  total  service  demanded 
of  the  universities  will  not  diminish.  On  the  contrary, 
it  will  continue  as  now  to  increase  in  response  to  grow- 
ing need.  The  case,  then,  is  clear:  the  number  of 
servants  must  be  increased,  the  number  of  those  who 
are  to  do  the  work  must  be  greatly  multiplied.  And 
thus  the  problem  becomes  a  financial  one.  But  a  uni- 
versity is  not  a  money-making  institution.  Its  function 
is  to  convert  the  physical  into  the  spiritual,  to  transform 
the  things  of  matter  into  the  things  of  mind.  It  has, 
however,  a  physical  body,  without  which  it  may  not 
dwell  among  men;  and,  for  the  support  of  it,  it  depends 
and  must  depend,  whether  through  legislative  appro- 
priation or  the  benefaction  of  individuals,  ultimately 
upon  the  people.  These  now  possess  the  means  in 
ample  measure,  and  the  promptings  of  generosity  are 
in  the  hearts  of  many  wealthy  and  sagacious  men. 

And  so  the  problem  revolves  upon  itself  and  once 
more  turns  full  upon  us  its  theoretic  aspect.  Its  solu- 
tion awaits  public  appreciation  of  its  significance  and 
its  terms.  It  is  above  all  else  a  question  of  enlighten- 
ment. Just  here,  if  I  am  not  mistaken,  is  the  measure- 
less opportunity  of  the  university  president.  Beyond 
all  others,  he  is  spokesman  and  representative  before 
the  people  of  their  highest  spiritual  interests.  Their 
ideals  and  aspirations  will  scarcely  surpass  his  own. 
The  problem  must  be  conceived  boldly  in  truth  and 
presented  in  its  larger  aspects.  It  must  be  seen  and  be 
felt  to  be  the  supreme  problem  of  our  civilisation.  As 
a  people  we  have  yet  to  learn  the  lesson  deeply  that 
research,  the  competent  application  in  any  field  what- 
ever of  human  interest  of  any  effective  method  whatever 
for  the  discovery  of  truth  and  enlarging  the  bounds  of 
knowledge,  is  the  highest  form  of  human  activity.  We 


RESEARCH   IN  AMERICAN   UNIVERSITIES  219 

have  yet  to  learn  that  a  nation,  a  state,  a  university 
without  investigators,  is  a  community  without  men  of 
profoundest  conviction.  For  this  can  not  be  gained  by 
conning  books;  it  can  not  be  inherited;  it  is  not  merely 
a  pious  hope  or  a  pleasing  superstition.  It  is  not  an 
obsession. 

As  Helmholz  has  said,  a  teacher  "who  desires  to 
give  his  hearers  a  perfect  conviction  of  the  truth  of  his 
principles  must,  first  of  all,  know  from  his  own  experi- 
ence how  conviction  is  acquired  and  how  not.  He 
must  have  known  how  to  acquire  conviction  where  no 
predecessor  had  been  before  him  —  that  is,  he  must 
have  worked  at  the  confines  of  knowledge  and  have  con- 
quered new  regions."  We  have  yet  to  learn  that  the 
value  of  a  university  professor  can  not  be  estimated  by 
counting  the  hours  he  stands  before  his  classes.  We 
have  yet  to  learn  to  prefer  standards  of  quality  to  units 
of  quantity.  We  have  yet  to  learn  that  the  spirit  of 
pure  research,  the  highest  productive  genius,  has  no 
direct  concern  whatever  with  the  useful;  that,  while 
it  does  without  intention  create  an  atmosphere  in  which 
utilities  most  greatly  flourish,  it  is  itself  concerned  solely 
with  the  true;  we  have  yet  to  learn  that  "the  action 
of  faculty  is  imperious  and  always  excludes  the  reflec- 
tion why  it  acts."  When  these  and  kindred  lessons 
shall  have  been  taken  to  heart,  our  emancipation,  now 
well  begun,  will  advance  towards  completion;  the  Amer- 
ican university  will  come  to  its  own;  and  our  present 
civilisation  will  speedily  pass  to  the  rank  of  the  highest 
and  best. 


PRINCIPIA  MATHEMATICA1 

MATHEMATICIANS,  many  philosophers,  logicians  and 
physicists,  and  a  large  number  of  other  people  are  aware 
of  the  fact  that  mathematical  activity,  like  the  activity 
in  numerous  other  fields  of  study  and  research,  has  been 
in  large  part  for  a  century  distinctively  and  increasingly 
critical.  Every  one  has  heard  of  a  critical  movement 
in  mathematics  and  of  certain  mathematicians  distin- 
guished for  their  insistence  upon  precision  and  logical 
cogency.  Under  the  influence  of  the  critical  spirit  of 
the  time  mathematicians,  having  inherited  the  tradi- 
tional belief  that  the  human  mind  can  know  some  propo- 
sitions to  be  true,  convinced  that  mathematics  may 
not  contain  any  false  propositions,  and  nevertheless 
finding  that  numerous  so-called  mathematical  proposi- 
tions were  certainly  not  true,  began  to  re-examine  the 
existing  body  of  what  was  called  mathematics  with  a 
view  to  purging  it  of  the  false  and  of  thus  putting  an 
end  to  what,  rightly  viewed,  was  a  kind  of  scientific 
scandal.  Their  aim  was  truth,  not  the  whole  truth, 
but  nothing  but  truth.  And  the  aim  was  consistent 
with  the  traditional  faith  which  they  inherited.  They 
believed  that  there  were  such  things  as  self-evident  prop- 
ositions, known  as  axioms.  They  believed  that  the 
traditional  logic,  come  down  from  Aristotle,  was  an 
absolutely  perfect  machinery  for  ascertaining  what  was 
involved  in  the  axioms.  At  this  stage,  therefore,  they 

1  An  account  of  Messrs.  Whitehead  and  Russell's  great  work  bearing 
this  title.  Printed  in  Science,  vol.  XXV. 


PRINCIPIA   MATHEMATICA  221 

believed  that,  in  order  that  a  given  branch  of  mathe- 
matics should  contain  truth  and  nothing  but  truth,  it 
was  sufficient  to  find  the  appropriate  axioms  and  then, 
by  the  engine  of  deductive  logic,  to  explicate  their  mean- 
ing or  content.  To  be  sure,  one  might  have  trouble 
to  "find"  the  axioms  and  in  the  matter  of  explication 
one  might  be  an  imperfect  engineer;  but  by  trying  hard 
enough  all  difficulties  could  be  surmounted  for  the 
axioms  existed  and  the  engine  was  perfect.  But  mathe- 
maticians were  destined  not  to  remain  long  in  this 
comfortable  position.  The  critical  demon  is  a  restless 
and  relentless  demon;  and,  having  brought  them  thus 
far,  it  soon  drove  them  far  beyond.  It  was  discovered 
that  an  axiom  of  a  given  set  could  be  replaced  by  its 
contradictory  and  that  the  consequences  of  the  new  set 
stood  all  the  experiential  tests  of  truth  just  as  well  as 
did  the  consequences  of  the  old  set,  that  is,  perfectly. 
Thus  belief  in  the  self-evidence  of  axioms  received  a 
fatal  blow.  For  why  regard  a  proposition  self-evident 
when  its  contradictory  would  work  just  as  well?  But 
if  we  do  not  know  that  our  axioms  are  true,  what  about 
their  consequences?  Logic  gives  us  these,  but  as  to 
their  being  true  or  false,  it  is  indifferent  and  silent. 

Thus  mathematics  has  acquired  a  certain  modesty. 
The  critical  mathematician  has  abandoned  the  search 
for  truth.  He  no  longer  flatters  himself  that  his  proposi- 
tions are  or  can  be  known  to  him  or  to  any  other  human 
being  to  be  true;  and  he  contents  himself  with  aiming 
at  the  correct,  or  the  consistent.  The  distinction  is 
not  annulled  nor  even  blurred  by  the  reflection  that 
consistency  contains  immanently  a  kind  of  truth.  He 
is  not  absolutely  certain,  but  he  believes  profoundly 
that  it  is  possible  to  find  various  sets  of  a  few  proposi- 
tions each  such  that  the  propositions  of  each  set  are 


222  PRINCIPIA   MATHEMATICA 

compatible,  that  the  propositions  of  such  a  set  imply 
other  propositions,  and  that  the  latter  can  be  deduced 
from  the  former  with  certainty.  That  is  to  say,  he 
believes  that  there  are  systems  of  coherent  or  consist- 
ent propositions,  and  he  regards  it  his  business  to  dis- 
cover such  systems.  Any  such  system  is  a  branch  of 
mathematics.  Any  branch  contains  two  sets  of  ideas 
(as  subject  matter,  a  third  set  of  ideas  are  used  but 
are  not  part  of  the  subject  matter)  and  two  sets  of 
propositions  (as  subject  matter,  a  third  set  being  used 
without  being  part  of  the  subject) :  that  is,  any  branch 
contains  a  set  of  ideas  that  are  adopted  without  defini- 
tion and  a  set  that  are  defined  in  terms  of  the  others; 
and  a  set  of  propositions  adopted  without  proof  and 
called  assumptions  or  principles  or  postulates  or  axioms 
(but  not  as  true  or  as  self-evident)  and  a  set  deduced 
from  the  former.  A  system  of  postulates  for  a  given 
branch  of  mathematics  —  a  variety  of  systems  may  be 
found  for  a  same  branch  — •  is  often  called  the  founda- 
tion of  that  branch.  And  that  is  what  the  layman 
should  think  when,  as  occasionally  happens,  he  meets 
an  allusion  to  the  foundation  of  the  theory  of  the  real 
variable,  or  to  the  foundation  of  Euclidean  geometry 
or  of  projective  geometry  or  of  Mengenlehre  or  of  some 
other  branch  of  mathematics.  The  founding,  in  the  sense 
indicated,  of  various  distinct  branches  of  mathematics 
is  one  of  the  great  outcomes  of  a  century  of  critical 
activity  in  the  science.  It  has  engaged  and  still  en- 
gages the  best  efforts  of  men  of  genius  and  men  of 
talent.  Such  activity  is  commonly  described  as  funda- 
mental. It  is  very  important,  but  fundamental  in  a 
strict  sense  it  is  not.  For  one  no  sooner  examines  the 
foundations  that  have  been  found  for  various  mathemat- 
ical branches  and  thereby  as  well  as  otherwise  gains 


PR1NC1PIA  MATHEMATICA  223 

a  deep  conviction  that  these  branches  are  constituents 
of  something  different  from  any  one  of  them  and  dif- 
ferent from  the  mere  sum  or  collection  of  all  of  them 
than  the  question  supervenes  whether  it  may  not  be 
possible  to  discover  a  foundation  for  mathematics  itself 
such  that  the  above-indicated  branch  foundations  would 
be  seen  to  be,  not  fundamental  to  the  science  itself, 
but  a  genuine  part  of  the  superstructure.  That  ques- 
tion and  the  attempt  to  answer  it  are  fundamental 
strictly.  The  question  was  forced  upon  mathematicians 
not  only  by  developments  within  the  traditional  field 
of  mathematics,  but  also  independently  from  develop- 
ments in  a  field  long  regarded  as  alien  to  mathematics, 
namely,  the  field  of  symbolic  logic.  The  emancipation 
of  logic  from  the  yoke  of  Aristotle  very  much  resembles 
the  emancipation  of  geometry  from  the  bondage  of 
Euclid;  and,  by  its  subsequent  growth  and  diversifica- 
tion, logic,  less  abundantly  perhaps  but  not  less  cer- 
tainly than  geometry,  has  illustrated  the  blessings  of 
freedom.  When  modern  logic  began  to  learn  from  such 
a  man  as  Leibnitz  (who  with  the  most  magnificent 
expectations  devoted  much  of  his  life  to  researches  in 
the  subject)  the  immense  advantage  of  the  systematic 
use  of  symbols,  it  soon  appeared  that  logic  could  state 
many  of  its  propositions  in  symbolic  form,  that  it  could 
prove  some  of  these,  and  that  the  demonstration  could 
be  conducted  and  expressed  in  the  language  of  symbols. 
Evidently  such  a  logic  looked  like  mathematics  and 
acted  like  it.  Why  not  call  it  mathematics?  Evidently 
it  differed  from  mathematics  in  neither  spirit  nor  form. 
If  it  differed  at  all,  it  was  in  respect  of  content.  But 
where  was  the  decree  that  the  content  of  mathematics 
should  be  restricted  to  this  or  that,  as  number  or  space? 
No  bne  could  find  it.  If  traditional  mathematics  could 


224  PRINCIPIA   MATHEMATICA 

state  and  prove  propositions  about  number  and  space, 
about  relations  of  numbers  and  of  space  configurations, 
about  classes  of  numbers  and  of  geometric  entities, 
modern  logic  began  to  prove  propositions  about  proposi- 
tions, relations  and  classes,  regardless  of  whether  such 
propositions,  relations  and  classes  have  to  do  with 
number  and  space  or  with  no  matter  what  other  spe- 
cific kind  of  subject.  At  the  same  time  what  was 
admittedly  mathematics  was  by  virtue  of  its  own  inner 
developments  transcending  its  traditional  limitations 
to  number  and  space.  The  situation  was  unmistakable: 
traditional  mathematics  began  to  look  like  a  genuine 
part  of  logic  and  no  longer  like  a  separate  something  to 
which  another  thing  called  logic  applied.  And  so  modern 
logicians  by  their  own  researches  were  forced  to  ask  a 
question,  which  under  a  thin  disguise  is  essentially  the 
same  as  that  propounded  by  the  bolder  ones  among  the 
critical  mathematicians,  namely,  is  it  not  possible  to 
discover  for  logic  a  foundation  that  will  at  the  same 
time  serve  as  a  foundation  for  mathematics  as  a  whole 
and  thus  render  unnecessary  (and  strictly  impossible) 
separate  foundations  for  separate  mathematical  branches? 
It  is  to  answer  that  great  question  that  Messrs. 
Whitehead  and  Russell  have  written  "Principia  Mathe- 
matica"  —  a  work  consisting  of  four  large  volumes, 
the  first  and  second  being  in  hand,  the  third  soon  to 
appear  —  and  the  answer  is  affirmative.  The  thesis 
is:  it  is  possible  to  discover  a  small  number  of  ideas 
(to  be  called  primitive  ideas)  such  that  all  the  other 
ideas  in  logic  (including  mathematics)  shall  be  defin- 
able in  terms  of  them,  and  a  small  number  of  propo- 
sitions (to  be  called  primitive  propositions)  such  that 
all  other  propositions  in  logic  (including  mathematics) 
can  be  demonstrated  by  means  of  them.  Of  course, 


PR1NC1PIA   MA  THEMATIC  A  335 

not  all  ideas  can  be  defined  —  some  must  be  assumed 
as  a  working  stock  —  and  those  called  primitive  are 
so  called  merely  because  they  are  taken  without  defini- 
tion; similarly  for  propositions,  not  all  can  be  proved, 
and  those  called  primitive  are  so  called  because  they  are 
assumed.  It  is  not  contended  by  the  authors  (as  it  was 
by  Leibnitz)  that  there  exist  ideas  and  propositions  that 
are  absolutely  primitive  in  a  metaphysical  sense  or  in 
the  nature  of  things;  nor  do  they  contend  that  but  one 
sufficient  set  of  primitives  (in  their  sense  of  the  term) 
can  be  discovered.  In  view  of  the  immeasurable  wealth 
of  ideas  and  propositions  that  enter  logic  and  mathe- 
matics, the  authors'  thesis  is  very  imposing;  and  their 
work  borrows  some  of  its  impressiveness  from  the  mag- 
nificence of  the  undertaking.  It  is  important  to  observe 
that  the  thesis  is  not  a  thesis  of  logic  or  of  mathematics, 
but  is  a  thesis  about  logic  and  mathematics.  It  can 
not  be  proved  syllogistically;  the  only  available  method 
is  that  by  which  one  proves  that  one  can  jump  through 
a  hoop,  namely,  by  actually  jumping  through  it.  If 
the  thesis  be  true,  the  only  way  to  establish  it  as  such 
is  to  produce  the  required  primitives  and  then  to  show 
their  adequacy  by  actually  erecting  upon  them  as  a 
basis  the  superstructure  of  logic  (and  mathematics)  to 
such  a  point  of  development  that  any  competent  judge  of 
such  architecture  will  say:  "Enough!  I  am  convinced. 
You  have  proved  your  thesis  by  actually  performing 
the  deed  that  the  thesis  asserts  to  be  possible." 

And  such  is  the  method  the  authors  have  employed. 
The  labor  involved  —  or  shall  we  call  it  austere  and 
exalted  play?  —  was  immense.  They  had  predecessors, 
including  themselves.  Among  their  earlier  works  Rus- 
sell's .  "Principles  of  Mathematics"  and  Whitehead's 
"Universal  Algebra"  are  known  to  many.  The  related 


226  PRINCIPIA   MATHEMATICA 

works  of  their  predecessors  and  contemporaries,  modern 
critical  mathematicians  and  modern  logicians,  Weier- 
strass,  Cantor,  Boole,  Peano,  Schroder,  Peirce  and  many 
others,  including  their  own  former  selves,  had  to  be 
digested,  assimilated  and  transcended.  All  this  was 
done,  in  the  course  of  more  than  a  score  of  years;  and 
the  work  before  us  is  a  noble  monument  to  the  authors' 
persistence,  energy,  acumen  and  idealism.  A  people 
capable  of  such  a  work  is  neither  crawling  on  its  belly 
nor  completely  saturated  with  commercialism  nor  wholly 
philistine.  There  are  preliminary  explanations  in  ordi- 
nary language  and  summaries  and  other  explanations 
are  given  in  ordinary  language  here  and  there  through- 
out the  book,  but  the  work  proper  is  all  in  symbolic 
form.  Theoretically  the  use  of  symbols  is  not  necessary. 
A  sufficiently  powerful  god  could  have  dispensed  with 
them,  but  unless  he  were  a  divine  spendthrift,  he 
would  not  have  done  so,  except  perhaps  for  the  reason 
that  whatever  is  feasible  should  be  done  at  least  once 
in  order  to  complete  the  possible  history  of  the  world. 
But  whilst  the  employment  of  symbols  is  theoretically 
dispensable,  it  is,  for-  man,  practically  indispensable. 
Many  of  the  results  in  the  work  before  us  could  not 
have  been  found  without  the  help  of  symbols,  and  even 
if  they  could  have  been  thus  found,  their  expression  in 
ordinary  speech,  besides  being  often  unintelligible,  owing 
to  complexity  and  involution,  would  have  required  at 
least  fifteen  large  volumes  instead  of  four.  Fortunately 
the  symbology  is  both  interesting  and  fairly  easy  to 
master.  The  difficulty  inheres  in  the  subject  itself. 

The  initial  chapter,  devoted  to  preliminary  explana- 
tions that  any  one  capable  of  nice  thinking  may  read 
with  pleasure  and  profit,  is  followed  by  a  chapter  of 
30  pages  dealing  with  "the  theory  of  logical  types." 


PRINCIPIA   MATHEMAT1CA  227 

Mr.  Russell  has  dealt  with  the  same  matter  in  volume 
30  of  the  American  Journal  of  Mathematics  (1908). 
One  may  or  may  not  judge  the  theory  to  be  sound  or 
adequate  or  necessary  and  yet  not  fail  to  find  in  the 
chapter  setting  it  forth  both  an  excellent  example  of 
analytic  and  constructive  thinking  and  a  worthy  model 
of  exposition.  The  theory,  which,  however,  is  recom- 
mended by  other  considerations,  originated  in  a  desire 
to  exclude  from  logic  automatically  by  means  of  its 
principles  what  are  called  illegitimate  totalities  and 
therewith  a  subtle  variety  of  contradiction  and  vicious 
circle  fallacy  that,  owing  their  presence  to  the  non- 
exclusion  of  such  totalities,  have  always  infected  logic 
and  justified  skepticism  as  to  the  ultimate  soundness 
of  all  discourse,  however  seemingly  rigorous.  (Such 
theoretic  skepticism  may  persist  anyhow,  on  other 
grounds.)  Perhaps  the  most  obvious  example  of  an 
illegitimate  totality  is  the  so-called  class  of  all  classes. 
Its  illegitimacy  may  be  shown  as  follows.  If  A  is  a 
class  (say  that  of  men)  and  £  is  a  member  of  it,  we 
say,  E  is  an  A.  Now  let  W  be  the  class  of  all  classes 
such  that  no  one  of  them  is  a  member  of  itself.  Then, 
whatever  class  x  may  be,  to  say  that  x  is  a  W  is  equiva- 
lent to  saying  that  x  is  not  an  x,  and  hence  to  say  that  W 
is  a  FT  is  equivalent  to  saying  that  W  is  not  a  Wl  Such 
illegitimate  totalities  (and  the  fallacies  they  breed)  are 
in  general  exceedingly  sly,  insinuating  themselves  under 
an  endless  variety  of  most  specious  disguises,  and  that, 
not  only  in  the  theory  of  classes  but  also  in  connection 
with  every  species  of  logical  subject-matter,  as  proposi- 
tions, relations  and  propositions!  functions.  As  the 
proposition  a!  function  —  any  expression  containing  a 
real  (as  distinguished  from  an  apparent)  variable  and 
yielding  either  non-sense  or  else  a  proposition  whenever 


228  PRINCIPIA   MATHEMATICA 

the  variable  is  replaced  by  a  constant  term  —  is  the 
basis  of  our  authors'  work,  their  theory  of  logical  types 
is  fundamentally  a  theory  of  types  of  prepositional 
functions.  It  can  not  be  set  forth  here  nor  in  fewer 
pages  than  the  authors  have  devoted  to  it.  Suffice  it  to 
say  that  the  theory  presents  prepositional  functions  as 
constituting  a  summitless  hierarchy  of  types  such  that 
the  functions  of  a  given  type  make  up  a  legitimate 
totality;  and  that,  in  the  light  of  the  theory,  truth  and 
falsehood  present  themselves  each  in  the  form  of  a 
systematic  ambiguity,  the  quality  of  being  true  (or 
false)  admitting  of  distinctions  in  respect  of  order,  level 
above  level,  without  a  summit.  When  Epimenides, 
the  Cretan,  says  that  all  statements  of  Cretans  are 
false,  and  you  reply  that  then  his  statement  is  false, 
the  significance  of  "false"  here  presents  two  orders  or 
levels;  and  logic  must  by  its  machinery  automatically 
prevent  the  possibility  of  confusing  them. 

Next  follows  a  chapter  of  20  pages,  which  all  phi- 
losophers, logicians  and  grammarians  ought  to  study, 
a  chapter  treating  of  Incomplete  Symbols  wherein  by 
ingenious  analysis  it  is  shown  that  the  ubiquitous  expres- 
sions of  the  form  "the  so  and  so"  (the  "the"  being 
singular,  as  "the  author  of  Waverley,"  "the  sine  of  a," 
"the  Athenian  who  drank  hemlock,"  etc.)  do  not  of 
themselves  denote  anything,  though  they  have  con- 
textual significance  essential  to  discourse,  essential  in 
particular  to  the  significance  of  identity,  which,  in  the 
world  of  discourse,  takes  the  form  of  "a  is  the  so  and 
so"  and  not  the  form  of  the  triviality,  a  is  a. 

After  the  introduction  of  88  pages,  we  reach  the  work 
proper  (so  far  as  it  is  contained  in  the  Volume  I.), 
namely,  Part  L:  Mathematical  Logic.  Here  enuncia- 
tion of  primitives  is  followed  by  series  after  series  of 


PRINC1PIA    MAI  HI  MA  I  1*  A 

theorems  and  demonstrations,  marching  through  578 
pages,  all  matter  being  clad  in  symbolic  garb,  except 
that  the  continuity  is  interrupted  here  and  there  by 
summaries  and  explanations  in  ordinary  language. 
Logic  it  is  called  and  logic  it  is,  the  logic  of  propositions 
and  functions  and  classes  and  relations,  by  far  the 
greatest  (not  merely  the  biggest)  logic  that  our  planet 
has  produced,  so  much  that  is  new  in  matter  and  in 
manner;  but  it  is  also  mathematics,  a  prolegomenon  to 
the  science,  yet  itself  mathematics  in  the  most  genuine 
sense,  differing  from  other  parts  of  the  science  only  in 
the  respects  that  it  surpasses  these  in  fundamentally, 
generality  and  precision,  and  lacks  traditionality.  Few 
will  read  it,  but  all  will  feel  its  effect,  for  behind  it  is 
the  urgence  and  push  of  a  magnificent  past:  two  thousand 
five  hundred  years  of  record  and  yet  longer  tradition  of 
human  endeavor  to  think  aright. 

Owing  to  the  vast  number,  the  great  variety  and  the 
mechanical  delicacy  of  the  symbols  employed,  errors 
of  type  are  not  entirely  avoidable  and  Volume  II.  opens 
with  a  rather  long  list  of  "errata  to  Volume  I."  The 
second  volume  is  composed  of  three  grand  divisions: 
Part  III.,  which  deals  with  cardinal  arithmetic;  Part  IV., 
which  is  devoted  to  what  is  called  relation-arithmetic; 
and  Part  V.,  which  treats  of  series.  The  theory  of  types, 
which  is  presented  in  Volume  I.,  is  very  important  in  the 
arithmetic  of  cardinals,  especially  in  the  matter  of 
existence-theorems,  and  for  the  convenience  of  the 
reader  Part  III.  is  prefaced  with  explanations  of  how 
this  theory  applies  to  the  matter  in  hand.  In  the  initial 
section  of  this  part  we  find  the  definition  and  logical 
properties  of  cardinal  numbers,  the  definition  of  car- 
dinal number  being  the  one  that  is  due  to  Frege,  namely, 
the  cardinal  number  of  a  class  C  is  the  class  of  all  classes 


230  PRINCIPIA   MATHEMAT1CA 

similar  to  C,  where  by  "similar"  is  meant  that  two 
classes  are  similar  when  and  only  when  the  elements 
of  either  can  be  associated  in  a  one-to-one  way  with 
the  elements  of  the  other.  This  section  consists  of 
seven  chapters  dealing  respectively  with  elementary 
properties  of  cardinals;  o  and  i  and  2;  cardinals  of 
assigned  types;  homogeneous  cardinals;  ascending 
cardinals;  descending  cardinals;  and  cardinals  of  rela- 
tional types.  Then  follows  a  section  treating  of  addi- 
tion, multiplication  and  exponentiation,  where  the 
logical  muse  handles  such  themes  as  the  arithmetical 
sum  of  two  classes  and  of  two  cardinals;  double  simi- 
larity; the  arithmetical  sum  of  a  class  of  classes;  the 
arithmetical  product  of  two  classes  and  of  two  cardinals; 
next,  of  a  class  of  classes;  multiplicative  classes  and 
arithmetical  classes;  exponentiation;  greater  and  less. 
Thus  no  less  than  186  large  symbolically  compacted 
pages  deal  with  properties  common  to  finite  and  infinite 
classes  and  to  the  corresponding  numbers.  Nevertheless 
finites  and  infinites  do  differ  in  many  important  re- 
spects, and  as  many  as  116  pages  are  required  to  present 
such  differences  under  such  captions  as  arithmetical 
substitution  and  uniform  formal  numbers;  subtraction; 
inductive  cardinals;  intervals;  progressions;  Aleph 
null,  Ko;  reflexive  classes  and  cardinals;  the  axiom  of 
infinity;  and  typically  indefinite  inductive  cardinals. 

As  indicating  the  fundamental  character  of  the  "Prin- 
cipia"  it  is  noteworthy  that  the  arithmetic  of  relations 
is  not  begun  earlier  than  page  301  of  the  second  huge 
volume.  In  this  division  the  subject  of  thought  is 
relations  including  relations  between  relations.  If  RI 
and  R-t  are  two  relations  and  if  F\  and  F2  are  their 
respective  fields  (composed  of  the  things  between  which 
the  relations  subsist),  it  may  happen  that  FI  and  F% 


PRINCIPIA   MATHEMATICA  23! 

can  be  so  correlated  that,  if  any  two  terms  of  FI  have 
the  relation  R\,  their  correlates  in  F2  have  the  relation 
Rtt  and  vice  versa.  If  such  is  the  case,  RI  and  RI  are 
said  to  be  like  or  to  be  ordinally  similar.  Likeness  of 
relations  is  analogous  to  similarity  of  classes,  and,  as 
cardinal  number  of  classes  is  defined  by  means  of  class 
similarity,  so  relation-number  of  relations  is  denned  by 
means  of  relation  likeness.  And  209  pages  are  devoted 
to  the  fundamentals  of  relation  arithmetic,  the  chief 
headings  of  the  treatment  being  ordinal  similarity  and 
relation-numbers;  internal  transformation  of  a  rela- 
tion; ordinal  similarity;  definition  and  elementary 
properties  of  relation-numbers;  the  relation-numbers, 
of,  2,  and  i,;  relation-numbers  of  assigned  types;  homo- 
geneous relation-numbers;  addition  of  relations  and  the 
product  of  two  relations;  the  sum  of  two  relations; 
addition  of  a  term  to  a  relation;  the  sum  of  the  rela- 
tions of  a  field;  relations  of  mutually  exclusive  rela- 
tions; double  likeness;  relations  of  relations  of  couples; 
the  product  of  two  relations;  the  multiplication  and 
exponentiation  of  relations;  and  so  on. 

The  last  259  pages  of  the  volume  deal  with  series.  A 
large  initial  section  is  concerned  with  such  properties 
as  are  common  to  all  series  whatsoever.  From  this 
exceedingly  high  and  tenuous  atmosphere,  the  reader  is 
conducted  to  the  level  of  sections,  segments,  stretches  and 
derivatives  of  series.  The  volume  closes  with  58  pages 
devoted  to  convergence,  and  the  limits  of  functions. 

To  judge  the  "Principia,"  as  some  are  wont  to  do, 
as  an  attempt  to  furnish  methods  for  developing  exist- 
ing branches  of  mathematics,  is  manifestly  unfair;  for 
it  is  no  such  attempt.  It  is  an  attempt  to  show  that 
the  entire  body  of  mathematical  doctrine  is  deducible 
from'  a  small  number  of  assumed  ideas  and  propositions. 


232  PRINCIPIA   MATHEMATICA 

As  such  it  is  a  most  important  contribution  to  the  theory 
of  the  unity  of  mathematics  and  of  the  compendence 
of  knowledge  in  general.  As  a  work  of  constructive 
criticism  it  has  never  been  surpassed.  To  every  one  and 
especially  to  philosophers  and  men  of  natural  science, 
it  is  an  amazing  revelation  of  how  the  familiar  terms 
with  which  they  deal  plunge  their  roots  far  into  the  dark- 
ness beneath  the  surface  of  common  sense.  It  is  a 
noble  monument  to  the  critical  spirit  of  science  and  to 
the  idealism  of  our  time. 


CONCERNING  MULTIPLE  INTERPRETATIONS 
OF  POSTULATE  SYSTEMS  AND  THE 
"EXISTENCE"  OF  HYPERSPACE l 

WHAT  do  we  mean  when  we  speak  of  n-dimensional 
space  and  n-dimensional  geometry,  where  n  is  greater 
than  3?  The  question  refers  to  talk  about  space  and 
geometry  that  are  n-dimensional  in  points,  for  ordinary 
space,  as  is  well  known,  is  4-dimensional  in  lines,  4-di- 
mensional  in  spheres,  5-dimensional  in  flat  line-pencils,  6- 
dimensional  in  circles,  etc.,  and  there  is  naturally  no 
mystery  involved  in  speaking  of  these  latter  varieties 
of  multi-dimensional  manifolds  and  their  geometries, 
no  matter  how  high  the  dimensionality  may  be.  No 
mystery  for  the  reason  that  in  these  geometries  every- 
thing Lies  within  the  domain  of  intuition  in  the  same 
sense  in  which  everything  in  ordinary  (point)  geometry 
lies  in  that  domain.  In  other  words,  these  n-dimen- 
sional geometries  are  nothing  but  theories  or  geometries 
of  ordinary  space,  that  arise  when  we  take  for  element, 
not  the  point,  but  some  other  entity,  as  the  line  or  the 
sphere,  .  .  .  whose  determination  in  ordinary  space 
requires  more  than  3  independent  data.  Of  these 
varieties  of  n-dimensional  geometry,  the  inventor  was 
Julius  PlUcker  (d.  1868),  but  Pliicker  declined  to  con- 
cern himself  with  spaces  and  geometries  of  more  than 
four  dimensions  in  points. 

1  Printed  in  The  Journal  of  Philosophy,  Psychology  and  ScUniiJic  Method, 
May  8,  1913. 


234    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

Since  Pliicker's  time,  however,  such  hyper-theories 
of  points  have  invaded  not  only  almost  every  branch 
of  pure  mathematics,  but  also  —  strangely  enough  — 
certain  branches  of  physical  science,  as,  for  example, 
the  kinetic  theory  of  gases.  As  to  the  manner  of  this 
latter  invasion  a  hint  may  be  instructive.  Given  N  gas 
molecules  enclosed,  say,  in  a  sphere.  These  molecules 
are,  it  is  supposed,  flying  about  hither  and  thither,  all 
of  them  in  motion.  Each  of  them  depends  on  six  co- 
ordinates, x,  y,  z,  u,  v,  w,  where  x,  y,  z,  are  the  usual 
positional  coordinates  of  the  molecule  regarded  as  a 
point  in  ordinary  space,  and  u,  v,  w  are  the  components 
of  the  molecule's  velocity  along  the  three  coordinate 
axes.  Knowing  the  six  things  about  a  given  molecule, 
we  know  where  it  is  and  the  direction  and  rate  of  its 
going.  The  N  molecules  making  up  the  gas  depend  on 
6N  coordinates.  At  any  instant  these  have  definite 
values.  These  values  together  define  the  "state"  of 
the  gas  at  that  instant.  Now  these  6N  values  are  said 
to  determine  a  point  in  space  of  6N  dimensions.  Thus 
is  set  up  a  one-one  correspondence  between  such  points 
and  the  varying  gas  states.  As  the  state  of  the  gas 
changes,  the  corresponding  point  generates  a  locus  in 
the  space  of  6N  dimensions.  In  this  way  the  behavior 
or  history  of  the  gas  gets  geometrically  represented  by 
loci  in  the  hyperspace  in  question. 

Is  such  geometric  ^-dimensional  phraseology  merely 
a  geometric  way  of  speaking  about  non-spatial  things? 
Even  if  there  exists  a  space,  Sn,  one  may  employ  the 
language  appropriate  to  the  geometry  of  the  space 
without  having  the  slightest  reference  to  it,  and,  indeed, 
without  knowing  or  even  enquiring  whether  it  exists. 
This  use  of  geometric  speech  in  discourse  about  non- 
spatial  things  is  not  only  possible,  but  in  fact  very  com- 


INTERPRETATIONS  OF  POSTULATE   SYSTEMS  235 

mon.  An  easily  accessible  example  of  it  may  be  found 
in  Bdcher1  where,  in  speaking  of  a  set  of  values  of  n  in- 
dependent variables  as  a  point  in  space  of  n  dimensions 
the  reader  is  told  that  the  author's  use  of  geometric 
language  for  the  expression  of  algebraic  facts  is  due  to 
certain  advantages  of  that  language  compared  with  the 
language  of  algebra  or  of  analysis;  he  is  told  that  the 
geometric  terms  will  be  employed  "in  a  wholly  conven- 
tional algebraic  sense"  and  that  "we  do  not  propose 
even  to  raise  the  question  whether  in  any  geometric 
sense  there  is  such  a  thing  as  space  of  more  than  three 
dimensions." 

It  is  held  by  many,  including  perhaps  the  majority 
of  mathematicians,  that  there  are  no  hyperspaces  of 
points  and  that  n-dimensional  geometries  are,  rightly 
speaking,  not  geometries  at  all,  but  that  the  facts 
dealt  with  in  such  so-called  geometries  are  nothing  but 
algebraic  or  analytic  or  numeric  facts  expressed  in 
geometric  language.  If  this  opinion  be  correct,  then 
the  extensive  and  growing  application  of  geometric 
language  to  analytical  theories  of  higher  dimensionality 
indicates  a  high  superiority  of  geometric  over  analytic 
speech,  and  it  becomes  a  problem  for  psychology  to 
ascertain  whether  the  mentioned  superiority  is  ade- 
quate to  explain  the  phenomenon  in  question  and,  if 
it  be  adequate,  to  show  wherein  the  superiority  resides. 

No  doubt  geometric  language  has  a  kind  of  esthetic 
value  that  is  lacking  in  the  speech  of  analysis,  for  the 
former,  being  transfused  with  the  rich  reminiscences 
of  sensibility,  constantly  awakens  a  delightful  sense, 
as  thinking  proceeds,  of  the  colors,  forms,  and  motions 
of  the  sensuous  world.  This  is  an  emotional  value.  No 
doubt,  too,  geometric  language  has,  in  its  distinctive 

1  "Introduction  to  Higher  Algebra,"  page  9. 


236    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

conciseness,  an  economic  superiority,  as  when,  for  ex- 
ample, one  speaks  of  the  points  of  the  4-dimensional 
sphere,  x2  +  yz  +  22  +  w2  =  r2,  instead  of  speaking  of  the 
various  systems  of  values  of  the  variables  x,  y,  z,  w  that 
satisfy  the  equation  x*  +  . . .  =  r2.  Additional  advan- 
tages of  geometric  over  analytic  speech  are  brought  to 
light  in  the  following  remarks  by  Poincare  in  his  ad- 
dress, "L'Avenir  des  Mathematiques "  (1908): 

"Un  grand  avantage  de  la  geometrie,  c'est  precisement 
que  les  sens  y  peuvent  venir  au  secours  de  1'intelligence, 
et  aident  a  deviner  la  route  a  suivre,  et  bien  des  esprits 
preferent  ramener  les  problemes  d'analyse  a  la  forme 
geometrique.  Malheureusement  nos  sens  ne  peuvent 
nous  mener  bien  loin,  et  ils  nous  faussent  compagnie 
des  que  nous  voulons  nous  envoler  en  dehors  des  trois 
dimensions  classiques.  Est-ce  a  dire  que,  sortis  de  ce 
domaine  restreint  ou  ils  semblent  vouloir  nous  enfermer, 
nous  ne  devons  plus  compter  que  sur  Fanalyse  pure  et 
que  toute  geometrie  a  plus  de  trois  dimensions  est  vaine 
et  sans  objet?  Dans  la  generation  qui  nous  a  precedes, 
les  plus  grands  maitres  auraient  repondu  'oui';  nous 
sommes  anjourd'hui  tellement  familiarises  avec  cette 
notion  que  nous  pouvons  en  parler,  meme  dans  un  cours 
d'universite,  sans  provoquer  trop  d'etonnement. 

"Mais  a  quoi  peut-elle  servir?  II  est  aise  de  le  voir: 
elle  nous  donne  d'abord  un  langage  tres  commode,  qui 
exprime  en  termes  tres  concis  ce  que  le  langage  analytique 
ordinaire  dirait  en  phrases  prolixes.  De  plus,  ce  langage 
nous  fait  nommer  du  meme  nom  ce  qui  se  ressemble  et 
affirme  des  analogies  qu'il  ne  nous  permet  plus  d'oublier. 
II  nous  permet  done  cenore  de  nous  diriger  dans  cet 
espace  qui  est  trop  grand  pour  nous  et  que  nous  ne 
pouvons  voir,  en  nous  rappelant  sans  cesse  1'espace 
visible  qui  n'en  est  qu'une  image  imparfaite  sans  doute, 


INTERPRETATIONS  OF  POSTULATE   SYSTEMS          237 

mais  que  en  est  encore  une  image.  Ici  encore,  comme 
dans  tous  les  exemples  pr£c6dents,  c'est  1'analogie  avec 
ce  qui  est  simple  qui  nous  permet  de  comprendre  ce  qui 
est  complexe." 

The  question  of  determining  the  comparative  advan- 
tages and  disadvantages  of  the  languages  of  geometry 
and  analysis  is  a  very  difficult  one.  It  is  evidently  in 
the  main  a  psychological  problem.  It  appears  that  no 
serious  and  systematic  attempt  has  ever  been  made  to 
solve  it.  Here,  it  seems,  is  an  inviting  opportunity  for 
a  properly  qualified  psychologist,  it  being  understood 
that  proper  qualification  would  include  a  familiar  knowl- 
edge of  the  languages  in  question.  The  interest  and 
manifold  utility  of  such  a  study  are  obvious.  In  the 
course  of  such  an  investigation  it  would  probably  be 
found  that  the  superiority  of  geometric  over  analytic 
speech  is  alone  sufficient  to  account  for  the  extensive 
and  rapidly  increasing  literature  of  what  is  called  n- 
dimensional  geometry  and  that,  in  order  to  account  for 
the  rise  of  such  literature,  it  is  therefore  not  necessary 
to  suppose  the  existence  of  n-dimensional  spaces,  Sm, 
the  facts  dealt  with  in  the  literature  being,  it  could  be 
supposed,  nothing  but  analytic  facts  expressed  in  geo- 
metric language. 

If  such  a  result  were  found,  would  it  follow  that  5, 
does  not  exist  and  that  consequently  n-dimensional  geom- 
etry must  be  nothing  but  analysis  in  geometric  garb? 
The  answer  is,  no;  for  we  may  and  we  often  do  assign 
an  adequate  cause  of  a  phenomenon  or  event  without 
assigning  the  actual  cause;  and  so  the  possibility  would 
remain  that  n-dimensional  geometry  has  an  appropriate 
object  or  subject,  namely,  a  space  5.,  which,  though 
without  sensuous  existence,  yet  has  every  kind  of  exist- 
ence that  may  warrantably  be  attributed  to  ordinary 


238     INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

geometric  space,  Ss.  For  this  last,  though  it  is  imitated 
by  (or  imitates)  sensible  space,  as  an  ideal  model  or 
pattern  is  imitated  by  (or  imitates)  an  imperfect  copy, 
it  is  not  identical  with  it.  83  is  not  tactile  space,  nor 
visual  space,  nor  that  of  muscular  sensation,  nor  the 
space  of  any  other  sense,  nor  of  all  the  senses  —  it  is 
a  conceptual  space;  and  whether  there  are  or  are  not 
spaces  $4,  85,  etc.,  which  have  every  sort  of  existence 
rightly  attributable  to  ordinary  geometric  space,  S3,  and 
which  differ  from  the  latter  only  in  the  accident  of 
dimensionality  and  in  the  further  accident  that  6*3  ap- 
pears in  the  r61e  of  an  ideal  prototype  for  an  actual 
sensible  space,  whilst  £4,  Ss,  etc.,  do  not  present  such 
an  appearance,  —  that  is  the  question  which  remains 
for  consideration. 

A  friend  called  at  my  study,  and,  finding  me  at  work, 
asked,  "What  are  you  doing?"  My  reply  was:  "I  am 
trying  to  tell  how  a  world  which  probably  does  not  exist 
would  look  if  it  did."  I  had  been  at  work  on  a  chapter 
of  what  is  called  4-dimensional  geometry.  The  incident 
occurred  ten  years  ago.  The  reply  to  my  friend  no 
longer  represents  my  conviction.  Subsequent  reflection 
has  convinced  me  that  a  space,  Sn,  of  four  or  more 
dimensions  has  every  kind  of  existence  that  may  be 
rightly  ascribed  to  the  space,  Ss,  of  ordinary  geometry. 

The  following  paragraphs  present  —  merely  in  out- 
line, for  space  is  lacking  for  a  minute  presentation  —  the 
considerations  that  have  led  me  to  the  conclusion  above 
stated. 

Let  sensible  space  be  denoted  by  s-S3.  We  know  that 
sS3  is  discontinuous  (in  the  mathematical  sense  of  the 
term)  and  that  it  is  irrational.  By  saying  that  it  is 
irrational  I  mean  what  common  experience  as  well  as 
the  results  of  experimental  psychology  prove:  that 


INTERPRETATIONS  OF  POSTULATE  SYSTEMS    239 

three  sensible  extensions  of  a  same  type,  let  us  for 
definiteness  say  three  sensible  lengths,  /i,  It,  /,,  may  be 
such  that 

(i)  /!  =  **,*»  =  /,,/!*/,. 

Because  sS3  is  thus  irrational,  because  it  is  radically 
infected  with  such  contradictions  as  (i),  this  space  is 
not,  and  can  not  be,  the  subject  or  object  of  geometry, 
for  geometry  is  rational;  it  does  not  admit  three  such 
extensions  as  those  in  (i).  Not  only  do  such  contra- 
dictions as  (i)  render  s$3  impossible  as  a  subject  or 
object  of  geometry,  but,  when  encountered,  they  pro- 
duce intolerable  intellectual  pain  —  nay,  if  they  could 
not  in  somewise  be  transcended  or  overcome,  they  would 
produce  intellectual  death,  for,  unless  the  law  of  non- 
contradiction be  preserved,  concatentative  thinking,  the 
life  of  intellect,  must  cease.  In  case  of  intellect  we  may 
say  that  its  struggle  for  existence  is  a  struggle  against 
contradictions.  But  mere  existence  is  not  the  character- 
istic aim  or  aspiration  of  intellect.  Its  aim,  its  aspira- 
tion, its  joy,  is  compatibility.  Indeed,  intellect  seems 
to  be  controlled  by  two  forces,  a  vis  a  tergo  and  a  vis 
a  fronte:  it  is  driven  by  discord  and  drawn  by  concord. 
Intellect  is  a  perpetual  suitor,  the  object  of  the  suit 
being  harmony,  the  beautiful  daughter  of  the  muses. 
Its  perpetual  enemy  is  the  immortal  demon  of  discord, 
ever  being  overcome,  but  never  vanquished. 

The  victory  of  intellect  over  the  characteristic  con- 
tradictions inherent  in  sS\  is  won  through  what  we  call 
conception.  That  is  to  say  that  either  we  find  or  else  we 
create  another  kind  of  space  which,  in  order  to  distin- 
guish it  nominally  and  symbolically  from  sS3,  we  may 
call  conceptual  space,  and  denote  by  cSt.  Unlike  sSi, 
cSt  is  mathematically  continuous  and  it  is  rational.  Like 
sS3.  cS3  is  extended,  it  has  room,  but  the  room  and 


240    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

the  extensions  are  not  sensible,  they  are  conceptual;  and 
these  extensions  are  such  that,  if  ll}  h,  k  be  three  amounts 
of  a  given  type  of  extension,  as  length,  say,  and  if  /i 
=  lz  and  lz  =  Is,  then  li  =  13.  The  space  cSs,  whether  we 
regard  it  as  found  by  the  intellect  or  as  created  by  it, 
is  the  subject  or  object  of  geometry.  The  current 
vulgar  confusion  of  sS3  and  cS3  is  doubtless  due  to  the 
fact  that  the  former  imitates  the  latter,  or  the  latter 
the  former,  as  a  sensible  thing  imitates  its  ideal,  or  as 
an  ideal  (of  a  sensible  thing)  may  be  said  to  imitate 
that  thing;  for  it  is  precisely  such  alternative  or  mutual 
imitation  that  enables  us  in  a  measure  to  control  the 
sensible  world  through  its  conceptual  counterpart;  and 
so  the  exigencies  of  practical  affairs  and  the  fact  that 
reciprocally  imitating  things  each  reminds  us  of  the 
other  cooperate  to  cause  the  sensible  and  the  ideal,  the 
perceptual  and  the  conceptual,  to  mingle  constantly 
and  to  become  confused  in  that  part  of  our  mental  life 
that  belongs  to  the  sensible  and  the  conceptual  worlds 
of  three  dimensions.  Nevertheless,  it  is  a  fact  to  be 
borne  in  mind  that  cS3  is  a  subject  or  object  of  geom- 
etry and  that  sS3  is  not. 

Now,  in  order  to  construct  the  geometry  in  question, 
we  start  with  a  suitable  system  of  postulates  or  axioms 
expressing  certain  relations  among  what  are  called  the 
elements  of  cS3.  These  postulates,  together  with  such 
propositions  as  are  deducible  from  them,  constitute  the 
geometry  of  cS3.  I  shall  call  it  pure  geometry,  for  a 
reason  to  be  given  later,  and  shall  denote  it  by  pG3. 
For  definiteness  let  us  refer  to  the  famous  and  familiar 
postulates  of  Hilbert.  Any  other  system  would  do  as 
well.  In  the  Hilbert  system,  the  elements  are  called 
points,  lines,  and  planes.  It  is  customary  and  just  to 
point  out  that  the  terms  point,  line,  and  plane  are  not 


INTERPRETATIONS   OF   POSTULATE   SYSTEMS  24! 

defined,  and  in  critical  commentary  it  is  customary  to 
add: 

(A)  That,  consequently,  these  terras  may  be  taken  to 
be  the  names  of  any  things  whatsoever  with  the  single 
restriction   that   the   things   must   satisfy   the   relations 
stated  by  the  postulates; 

(B)  That,  when  some  admissible  or  possible  interpre- 
tation /  has  been  given  to  the  element-names,  the  postu- 
lates P  together  with  their  deducible  consequences  C 
constitute  a  definite  theory  or  doctrine  D; 

(C)  That  replacing  7  by  a  different  interpretation  /' 
produces  no  change  whatever  in  D; 

(D)  That    this   invariant   D   is    Euclidean    geometry 
of  three  dimensions;    and 

(£)  That,  if  we  are  to  speak  of  D  as  a  theory  or  geom- 
etry of  a  space,  this  space  is  nothing  but  the  ensemble 
of  any  kind  of  things  that  may  serve  for  an  interpreta- 
tion of  P. 

That  the  view  expressed  in  that  so-called  "critical 
commentary"  does  not  agree  with  common  sense  or 
with  traditional  usage  is  obvious.  That  it  will  not  bear 
critical  reflection  can,  I  believe,  be  made  evident.  Let 
us  examine  it  a  little.  In  order  to  avoid  the  prejudicial 
associations  of  the  terms  point,  line,  and  plane,  we  may 
replace  them  by  the  terms  "roint,"  "rine,"  and  "rane," 
so  that  the  first  postulate,  or  axiom,  as  Hilbert  calls  it, 
will  read:  Two  distinct  roints  always  completely  deter- 
mine  a  rine.  Or,  better  still,  we  may  replace  them  by 
the  symbols  e\t  c*,  e\,  so  that  the  reading  will  be:  Two 
distinct  e\s  always  completely  determine  an  4;  and  sim- 
ilarly for  the  remaining  postulates. 

We  will  suppose  the  phrasing  of  (A),  (B),  (C),  (D), 
(£),•  slightly  changed  to  agree  with  the  indicated  new 
phrasing  of  the  postulates. 


242     INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

It  seems  very  probable  that  there  are  no  termless 
relations,  i.  e.,  relations  that  do  not  relate.  It  seems 
very  probable  that  a  relation  to  be  a  relation  must  be 
something  actually  connecting  or  subsisting  between 
at  least  two  things  or  terms.  A  postulate  expressing 
a  relation  having  terms  is  at  all  events  ostensibly  a 
statement  about  the  terms,  and  so  it  would  seem  that, 
if  the  relation  be  supposed  to  be  termless,  the  statement 
ceases  to  be  a  statement  about  something  and,  in  so 
ceasing,  ceases  to  be  a  statement  that  is  true  or  else 
is  false.  In  discourse,  it  is  true,  there  is  frequent  seem- 
ing evidence  that  relations  are  often  thought  of  as 
termless,  as  when,  for  example,  we  speak  of  "a  relation 
and  its  terms";  but  then  we  speak  also  of  a  neckless 
fiddle  without  intending  to  imply  by  such  locution 
that  there  can  be  a  fiddle  without  a  neck.  As,  however, 
we  do  not  wish  the  validity  of  the  following  criticism  to 
depend  on  the  denial  of  the  possibility  of  termless  rela- 
tions, the  discussion  will  be  conducted  in  turn  under 
each  of  the  alternative  hypotheses:  (hi)  There  are  term- 
less relations;  (hz)  There  are  no  termless  relations.  We 
will  begin  with 

HYPOTHESIS  fa 

To  (A)  we  make  no  objection. 

Let  us  now  suppose  given  to  P  some  definite  inter- 
pretation /.  Let  us  grant  that  we  now  have  a  definite 
doctrine  D,  consisting  of  P  and  C.  Either  the  things 
which  in  I  the  e's  denote  have  or  they  have  not  con- 
tent, character,  or  meaning,  m,  in  excess  of  the  fact 
that  they  satisfy  P. 

(i)  Suppose  they  have  not  an  excessive  meaning  m. 
Denote  the  interpretation  by  /i  and  the  doctrine  by 
D\.  This  DI  is  a  queer  doctrine.  We  may  ask;  what 


INTERPRETATIONS   OF  POSTULATE   SYSTEMS  243 

does  D\  relate  or  refer  to?  That  is,  what  is  it  a  doctrine 
of  or  about?  The  question  seems  to  admit  of  no  intel- 
ligent or  intelligible  answer.  For  if  the  doctrine  is 
about  something,  it  is,  it  seems  natural  to  say,  a  doc- 
trine about  the  /i-things  (denoted  by  the  e's);  but, 
by  (i),  these  /i- things  can  not  be  characterized  or  in- 
dicated otherwise  than  by  the  fact  of  their  satisfying 
P;  and  so  it  appears  that  such  attempted  natural  an- 
swer is  reducible  and  equivalent  to  saying  (a)  that  the 
doctrine  D\  is  about  the  things  which  it  is  about.  In 
order  not  to  be  thus  defeated,  one  might  try  to  give  an 
informing  answer  by  saying  that  D\  is  a  doctrine,  not 
about  the  /i- things,  i.  c.,  not  about  terms  of  relations, 
but  about  the  relations  themselves.  Such  an  answer  is 
suspicious  on  account  of  its  unnaturalness,  and  it  is 
unnatural  because  the  propositions  of  D\  wear  the  ap- 
pearance of  talking  explicitly,  not  about  relations,  but 
about  terms  of  relations.  Moreover,  the  answer  is  not 
an  informing  one  unless  the  relations  that  the  doctrine 
D\  is  alleged  to  be  about  can  be  characterized  otherwise 
than  by  the  fact  of  their  being  satisfied  by  the  /i-things, 
for,  if  they  can  not  be  otherwise  characterized,  evidently 
by  (i)  the  answer  reduces  to  a  form  essentially  like 
that  of  (a).  May  not  one  escape  by  saying  that  the 
relations  which  D\  is  alleged  to  be  a  doctrine  about  are 
just  the  relations  expressed  by  the  propositions  in  D\? 
Does  this  attempted  characterization  make  the  answer 
in  question  an  informing  one?  If  D\  is  a  doctrine  about 
the  relations  expressed  by  its  propositions,  then  D\  says 
or  teaches  something  about  these  relations,  for  every 
doctrine,  if  it  be  about  something,  must  teach  or  say 
something  about  that  which  it  is  about.  In  the  case 
supposed,  what  does  D\  teach  about  the  relations? 
Nothing  except  that  they  are  satisfied  by  the  /i-things. 


244    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

In  other  words,  what  D\  teaches  about  the  relations 
expressed  by  its  propositions  is,  by  (i),  that  these  are 
satisfied  by  things  that  satisfy  them  —  a  not  very 
nutritious  lesson.  It  is  possible  to  make  a  yet  further 
attempt  so  to  indicate  the  relations  as  to  render  the 
answer,  that  the  doctrine  D\  is  about  relations,  an  in- 
forming one.  It  is  known  that  P  may  receive  an  inter- 
pretation /'  different  from  I\  in  that  the  /'-things  do 
not  satisfy  (i),  but  have  an  excessive  content,  character, 
or  meaning  m.  May  we  not  give  the  required  indica- 
tion of  the  relations  that  D\  is  said  to  be  a  doctrine 
about  by  saying  that  they  are  relations  satisfied  by  the 
/'-things,  the  presence  of  the  m  involved  making  the 
indication  genuine  or  effective?  It  seems  so  at  first. 
But  if  again  we  ask  what  D\  teaches  about  the  relations 
thus  indicated,  we  are  led  into  the  same  difficulty  as 
above.  Moreover,  when  we  ask  what  D\  is  a  doctrine 
about  we  expect  an  answer  in  terms  in  somewise  men- 
tioned or  intimated  in  the  D\  discourse,  whilst  in  the 
case  in  hand  the  required  indication  has  depended  on 
m,  a  thing  expressly  excluded  from  the  D\  discourse 
by  (i). 

So,  I  repeat,  D\  is  a  queer  doctrine. 

It  must  be  added  that  if  there  be  an  interpretation  /i, 
it  is  unique  of  its  kind;  for  if  I\  were  an  interpreta- 
tion satisfying  (i),  the  /I'-things  would  have  no  excess- 
ive meaning  m;  hence  they  would  be  simply  the  /i- 
things,  and  I\  and  I\  would  be  merely  two  symbols 
for  a  same  interpretation. 

Accordingly,  if  there  were  an  interpretation  /i,  but  no 
other,  i.  e.,  no  interpretation  /  in  which  the  /-things 
did  not  satisfy  (i),  then  (C)  would  be  pointless;  by  (D), 
DI  would  be  Euclidean  geometry  of  three  dimensions; 
and,  by  (£),  Euclidean  space,  if  we  wished  to  speak  of 


INTERPRETATIONS   OF   POSTULATE   SYSTEMS  245 

D\  as  a  geometry  of  a  space,  would  be  the  ensemble  of 
the  /i-things;  but,  if  we  wished  to  characterize  the  I\- 
things,  the  elements  of  Euclidean  space,  we  could  only 
say  that  they  are  the  things  satisfying  certain  relations, 
and,  if  we  wished  to  indicate  what  relations,  we  could 
only  say,  the  relations  satisfied  by  those  things:  a  very 
handsome  circle. 

In  the  following  it  will  be  seen  that  we  are  in  fact  not 
imprisoned  within  that  circle. 

(2)  Suppose  the  /-things  of  the  above-assumed  inter- 
pretation 7  do  not  satisfy  (i),  but  have  an  excessive 
meaning  m.  (It  is  known  that  such  an  7  is  possible, 
an  example  being  found  by  taking  for  an  e\  any  ordered 
triad  of  real  numbers  (*,  y,  z);  for  an  e\  the  ensemble 
of  triads  satisfying  any  two  distinct  equations, 


Biy+Ciz+  D!  =  o,  A&+  B*y+C*+  A  -o, 

in  neither  of  which  the  coefficients  are  all  of  them  zero; 
and  for  an  e\  the  ensemble  of  triads  satisfying  any  one 
such  equation;  the  presence  of  m  being  evident  in  count- 
less facts  such  as  the  fact,  for  example,  that  an  e\  is 
composed  of  numbers  studied  by  school-boys  or  useful 
in  trade  without  regard  to  their  ordered  triadic  rela- 
tionship.) Denote  the  assumed  definite  interpretation 
/  by  /i  to  remind  us  that  it  satisfies  (2),  and  denote 
the  corresponding  doctrine  by  D\.  It  is  immediately 
evident  that  there  is  an  interpretation  I\  and  hence  a 
doctrine  D\,  for  to  obtain  /i  it  is  sufficient  to  abstract 
from  the  m  of  the  /j-things  and  to  take  the  abstracts 
(which  plainly  satisfy  (i))  for  Ii-things. 

Are  Di  and  A  but  two  different  symbols  for  one  and 
the  same  doctrine,  as  asserted  by  (C)?  Evidently  not. 
For,  in  respect  of  DI,  we  can  give  an  informing  answer 
to  the  question,  what  is  A  a  doctrine  about?  Owing 


246    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

to  the  presence  of  the  m  in  the  /2-things,  the  answer 
will  be  an  informing  one  whether  it  be  the  natural  answer 
that  A  is  a  doctrine  about  the  /2-things,  or  one  of  the 
less  natural  answers,  that  D2  is  about  the  relations  having 
the  /2-things  for  terms,  that  D2  is  about  the  relations 
expressed  by  its  propositions;  whilst,  as  we  have  seen, 
owing  to  the  absence  of  m  in  the  /i-things  no  such 
answers  were,  in  respect  of  D\,  informing  answers. 

Can  not  (C)  be  saved  by  refusing  to  admit  that  there 
is  an  interpretation  /i,  and  so  refusing  to  admit  that 
there  is  a  Z>i?  If  there  is  no  I\  and  hence  no  D\,  then 
(C)  is  pointless  unless  there  is  an  /a'  and  so  a  ZY  in 
which  the  /2'-things  have  an  m'  different  from  the  m 
of  the  /2-things.  But  if  there  is  an  /2'  thus  different 
from  /2,  then  obviously  /Y  and  Dz  are,  contrary  to  (C), 
different  doctrines,  for  they  are  respectively  doctrines 
about  the  /2-things  and  the  /z'-things,  and  these  thing- 
systems  are  different  by  virtue  of  the  difference  of  m 
and  m'.  Now,  it  is  known  that  there  are  two  such  differ- 
ing interpretations  /2  and  /2'.  For  we  may  suppose  /2 
to  be  the  possible  interpretation  indicated  in  the  above 
parenthesis.  And  for  -/2'  we  may  take  for  e\  any  ordered 
triad  of  real  numbers,  except  a  specified  triad  (a,  b,  c), 
and  including  (  »  f  o>  t  »  )  ;  f  or  ^  the  ensemble  of  triads, 
except  (a,  b,  c),  that  satisfy  any  pair  of  equations, 

)  +  2Bi(x  -  a)  +  2Ci(y  -  b) 
+  2Di(*  -c)-  Ai(a*+  £2+  c1)  =  o, 

2)  +  *Bi(x  -  a)  +  iCt(y  -  b) 
+  2D2(z  -c)-  Ai(a*+  b*+c*)  =  o; 


and  for  et  the  ensemble  satisfying  any  one  such  equation. 
Just  as  when  we  compared  D\  and  D2,  so  here  the  con- 
clusion is,  that  (C)  is  not  valid. 


INTERPRETATIONS  OF  POSTULATE  SYSTEMS     247 

As  a  matter  of  fact  mathematicians  know  that  there 
are  possible  infinitely  many  different  interpretations  of 
P.  It  follows  from  the  foregoing  that  there  are  corre- 
spondingly many  different  doctrines.  For  the  sake  of 
completeness  we  may  include  D\  among  these,  although, 
for  the  purpose  of  answering  a  hypothetical  objec- 
tion, we  momentarily  supposed  D\  to  be  disputable  or 
inadmissible. 

Which  one  of  the  D's  is  (or  should  be  called)  Euclidean 
geometry  of  three  dimensions?  I  say  which  "one"? 
For,  as  no  two  are  identical,  it  would  be  willful  courting 
of  ambiguity  to  allow  that  two  or  more  of  them  should 
be  so  denominated.  Which  one,  then?  Evidently  not 
one  of  the  numerical  ones,  such,  for  example,  as  the 
two  above  specified.  For  who  has  ever  really  believed 
that  a  point,  for  example,  is  a  triad  of  numbers?  We 
know  that  the  Greeks  did  arrive  at  geometry;  we  know 
that  they  did  not  arrive  at  it  through  numbers;  and 
we  know  that,  in  their  thought,  points  were  not  number 
triads,  nor  were  planes  and  lines,  for  them,  certain 
ensembles  of  such  triads.  The  confusion,  if  anybody 
ever  was  really  thus  confused,  is  due  to  the  modern 
discovery  that  number  triads  and  certain  ensembles  of 
them  happen  to  satisfy  the  same  relations  as  the  Greeks 
found  to  be  satisfied  by  what  they  called  points,  lines, 
and  planes.  There  is  really  no  excuse  for  the  confusion, 
for,  if  Smith  is  taller  than  Brown,  and  yonder  oak  is 
taller  than  yonder  beech,  it  obviously  does  not  follow 
that  Smith  is  the  oak  and  Brown  the  beech. 

Evidently  Euclidean  geometry  of  three  dimensions 
is  that  particular  D  for  which  the  /-things  are  points, 
lines,  and  planes.  Here  it  is  certain  to  be  asked: 
What,  then,  are  points,  lines,  and  planes?  And  the 
asker  will  mean  to  imply  that,  in  order  to  maintain  the 


248    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

proposition,  it  is  necessary  to  define  these  terms.  The 
proper  reply  is  that  is  it  not  necessary  to  define  them. 
All  that  can  be  reasonably  required  is  that  they  be 
indicated,  pointed  out,  sufficiently  described  for  purposes 
of  recognition,  for  what  we  desire  is  to  be  able  to  say 
or  to  recognize  what  Euclidean  geometry  is  about.  To 
the  question  one  might,  not  foolishly,  reply  that  the 
terms  in  question  denote  things  that  you  and  I,  if  we 
have  been  disciplined  in  geometry,  converse  under- 
standingly  about  when  we  converse  about  geometry, 
though  neither  of  us  is  able  to  say  with  absolute  pre- 
cision what  the  terms  mean.  For  who  does  not  know 
that  it  is  possible  to  write  an  intelligent  and  intelligible 
discourse  about  cats,  for  example,  without  being  able 
to  tell  (for  who  can  tell?)  precisely  what  a  cat  is?  And 
if  it  be  asked  what  the  discourse  is  about,  who  does 
not  know  that  it  is  an  informing  answer  to  say  that 
it  is  about  cats?  It  is  informing  because  the  term  cat 
has  an  excessive  meaning,  a  meaning  beyond  that  of 
satisfying  the  propositions  (or  relations)  of  the  discourse. 
Just  here  it  is  well  worth  while  to  point  out  an  im- 
portant lesson  in  the  procedure  of  Euclid.  Against 
Euclid  it  is  often  held  as  a  reproach  that  he  attempted 
to  define  the  element-names,  point,  line,  and  plane, 
since  no  definitions  of  them  could  render  any  logical 
service,  that  is,  in  the  strictly  deductive  part  of  the 
discourse.  But  to  render  no  logical  service  is  not  to 
render  no  service.  And  the  lesson  is  that  the  definitions 
in  question,  which  it  were  perhaps  better  to  call  de- 
scriptions, do  render  an  extralogical  service.  They 
render  such  service  not  only  in  guiding  the  imagination 
in  the  matter  of  invention,  but  also  in  serving  to  indi- 
cate, with  a  goodly  degree  of  success,  the  excessive 
meaning  m  of  the  elements  denoted  by  the  terms  in 


INTERPRETATIONS  OP  POSTULATE  SYSTEMS    249 

question  and  in  thus  serving  to  make  known  what  it  is 
that  the  deductive  part  of  the  discourse  is  about.  One 
should  not  forget  that  no  discourse,  no  doctrine,  not 
even  so-called  pure  logic  itself,  is  exclusively  deductive, 
for  in  any  doctrine  there  is  reference,  implicit  or  ex- 
plicit, to  something  extradeductive  or  extralogical, 
reference,  that  is,  to  something  which  the  doctrine  is 
about. 

Are  the  three  Euclidean  "definitions,"  thus  viewed 
as  descriptions,  sufficient  or  adequate  to  the  service 
that  they  are  here  viewed  as  rendering?  If  by  suffi- 
cient or  adequate  be  meant  exhaustive,  the  answer  is, 
of  course,  no.  For  we  may  confidently  say  that  no 
possible  description,  that  is,  no  description  involving 
only  a  finite  number  of  words,  can  exhaust  the  meaning 
of  a  system  of  terms  except,  possibly,  in  the  special 
case  where  these  have  no  meaning  beyond  what  they 
must  have  in  order  merely  to  satisfy  a  finite  number  of 
postulates.  But  exhaustive  is  not  what  is  meant  by 
adequate.  To  employ  a  previous  illustration,  it  is  not 
necessary  to  give  or  to  attempt  an  exhaustive  descrip- 
tion of  "cat"  in  order  to  tell  adequately  what  it  is 
that  a  discourse  ostensibly  about  cats  is  ostensibly  about. 
It  is  a  question  of  intent.  A  description  is  nearly,  if 
not  quite,  adequate  if  it  enables  us  to  avoid  thinking 
that  terms  are  intended  to  denote  what  they  are  not 
intended  to  denote.  And,  whilst  we  may  not  admit 
that  the  three  Euclidean  "descriptions"  are  the  best  that 
can  be  invented  for  the  purpose,  yet  we  must  allow 
that  they  have  long  served  the  end  in  question  pretty 
effectively  and  that  they  are  qualified  to  continue  such 
service.  They  have  been  and  they  are  good  enough, 
for  example,  to  save  us  from  thinking  that  the  things 
which  in  geometry  have  been  denoted  by  the  terms, 


250    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

points,  lines,  and  planes,  are  identical  with  number 
triads,  etc.  The  open  secret  of  their  thus  saving  us  is 
no  doubt  in  their  causing  us  to  think  of  points,  lines, 
and  planes  in  terms  of,  or  in  essential  connection  with, 
what  we  know  as  extension,  whilst  numbers  and  number 
ensembles  are  not  things  naturally  so  conceived.  For 
evidently  the  notions  of  "length"  and  "breadth"  in- 
volved in  the  Euclidean  "descriptions"  are  not  metric 
in  meaning;  they  do  not  signify  definite  or  numeric 
quantities  or  amounts  of  something  (as  when  we  say 
the  length  of  this  or  that  thing  is  so  and  so  much); 
but  plainly  they  are  generic  notions  connoting  extension. 
It  is  safe  to  say  that  a  mind  devoid  of  the  concept  or 
the  sense  of  extension  could  not  know  what  things  the 
"descriptions"  aim  at  describing.  It  is  true  that 
Euclid's  "description"  of  a  point  as  "that  which  has 
no  part"  implies  a  denial  of  extension,  but  the  denial 
is  one  of  extension,  and,  in  its  contextual  atmosphere, 
it  is  felt  to  be  essential  to  an  adequate  indication  of 
what  is  meant  by  point.  On  the  other  hand,  if  one 
were  (and  how  unnatural  it  would  be!)  to  describe  an 
ordered  triad  of  numbers  as  "that  which  has  no  part," 
it  would  be  immediately  necessary  to  explain  away  the 
seeming  falsity  of  the  description  by  saying  that  the 
triad  is  not  the  ordered  multiplicity  (of  three  numbers) 
as  a  multiplicity,  but  is  merely  the  uniphase  of  the 
multiplicity,  and  that  it  is  this  uniphase  which  has  no 
part.  If,  next,  we  were  to  say  that  thus  extension  is 
denied  to  the  uniphase,  the  statement,  though  true, 
would  be  felt  to  be  inessential  to  an  adequate  indica- 
tion of  what  is  meant  by  a  triad  of  numbers.  Such 
felt  difference  is  alone  sufficient  to  make  any  one  pause 
who  is  disposed  to  adopt  the  current  creed  that  a  point 
is  nothing  but  an  ordered  triad  of  numbers.  It  is  not 


INTERPRETATIONS   OF  POSTULATE   SYSTEMS  251 

contended  that  a  point  is  composed  of  extension;  the 
contention  is  that  point  and  extension  are  so  connected 
that  a  mind  devoid  of  the  latter  notion  would  be  devoid 
of  the  former,  just  as  a  mind  devoid  of  the  notion  of 
variable  or  variation  would  be  devoid  of  the  notion 
of  constant,  though  a  constant  is  not  a  thing  consisting 
of  variation;  just  as  the  notion  of  limit  would  not  be 
intelligible  except  for  the  notion  of  something  that  may 
have  a  limit,  though  the  limit  is  not  composed  of  it; 
and  just  as  an  instant,  which  is  not  composed  of  time, 
would  not  be  intelligible  except  for  the  notion  of  time. 
In  a  discussion  of  such  matters  it  is  foolish  and  futile 
to  talk  about  "proofs."  The  question,  as  said,  is  one 
of  intent;  it  is  a  question  of  self -veracity,  of  getting 
aware  of  and  owning  what  it  is  that  we  mean  by  the 
terms  and  symbols  of  our  discourse.  If,  despite  the 
Euclidean  "descriptions"  and  despite  any  and  all  others 
that  may  supplement  or  supplace  them,  one  fails  to  see 
that  extension  is  essentially  involved  in  the  meaning  that 
the  terms  points,  lines,  and  planes,  are  intended  to  have, 
the  failure  will  be  because  "as  the  eyes  of  bats  are 
to  the  blaze  of  day,  so  is  the  reason  in  our  soul  to  the 
things  which  are  by  nature  most  evident  of  all."  Noth- 
ing is  more  evident  than  that  there  is  something  that 
is  called  extension.  We  have  but  to  open  our  eyes  to 
get  aware  that  we  are  beholding  an  expanse,  something 
extended.  We  see  things  as  extended:  things  as  ex- 
tended are  revealed  to  the  tactile  sense;  a  region  or 
room  involving  extension  is  a  datum  of  the  muscular 
sensations  connected  with  our  bodily  movements; 
and  so  on.  So  much  is  certain.  But  it  is  said  and 
rightly  said  that  these  are  sensible  things;  that  the 
extension  they  are  revealed  as  having  is  sensibU  exten- 
sion; that  these  sensibles  are  infected  with  contra- 


252    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

diction,  above  noted,  revealed  in  common  experience, 
and  confirmed  by  the  psychophysical  law  of  Weber  and 
Fechner;  that  geometry  is  free  from  contradiction; 
that,  therefore,  geometry  is  not  a  doctrine  about  these 
sensibles;  that  among  these  sensibles  are  not  the  things 
which  in  geometry  are  denoted  by  the  terms  point, 
line,  and  plane;  and  that,  if  these  terms  imply  or  con- 
note extension,  as  asserted,  this  extension  is  not  sensible 
extension.  Granted.  The  "connoted  extension"  is  not 
sensible,  it  is  conceptual.  How  know,  however,  that 
there  is  conceptual  extension?  The  answer  is,  by 
arriving  at  it.  (We  need  not  here  debate  whether  such 
"arriving"  is  best  called  creating  or  is  best  called  find- 
ing.) But  how  does  the  mind  arrive  at  it?  By  doing 
certain  things  to  the  sensibles,  the  raw  material  of  mental 
architecture.  What  things?  An  exhaustive  answer  is 
unnecessary  —  perhaps  impossible.  The  things  are  of 
two  sorts:  the  mind  gives  to  the  sensibles;  it  takes 
away  from  them.  Consider  for  example  a  sensible 
line.  From  it  the  conceptualizing  intellect  takes  away 
(abstracts  from,  disregards)  certain  things  that  the 
sensible  in  question  has  or  may  have,  as  color,  weight, 
temperature,  etc.,  including  part  of  the  extension,  thus, 
I  mean,  narrowing  and  thinning  away  all  breadth  and 
thickness.  What  of  the  extension  called  length?  Have 
the  narrowing  and  thinning  taken  it  away?  It  was  not 
so  intended,  the  opposite  was  intended.  Yet  no  sensible 
length  (extension)  remains.  Does  the  narrowing  and 
thinning  involve  shortening?  We  are  absolutely  certain 
that  it  does  not.  What,  then,  is  it  that  has  happened? 
Evidently  that,  by  the  indicated  taking  away,  the  mind 
has  arrived  at  insensible  length,  one  kind  of  insensible 
extension,  that  is,  at  conceptual  length,  one  kind  of 
conceptual  extension.  A  stretch,  we  are  sure,  remains, 


INTERPRETATIONS  OF  POSTULATE   SYSTEMS          253 

but  it  is  not  a  sensible  stretch.  The  extension  thus 
arrived  at  is  yet  not  the  extension  connoted  by  or  in- 
volved in  the  things  that  geometry  is  about,  for  in  the 
taking-away  process  of  arriving  at  it  there  is  nothing 
to  disinfect  it  of  the  contradictions  inherent  in  the 
sensible  with  which  we  started.  It  remains,  then,  to 
follow  the  indicated  process  of  taking  away  by  a  process 
of  giving,  that  is  to  say,  it  remains  to  endow  the  con- 
ceptual extension  (arrived  at)  with  continuity  so  as  to 
render  it  free  from  the  mentioned  contradictions.  This 
done,  the  kind  of  extension  meant  in  ordinary  geom- 
etry or  ordinary  geometric  space  is  arrived  at.  Such 
is,  in  kind,  the  conceptual  extension  that,  it  is  here 
held,  is  essential  to  what  the  geometric  terms,  point, 
line,  plane,  are  intended  to  mean.  Without  further 
talk  we  may  say  that  such  extension  is  essential  in  the 
conceptual  space  that,  we  may  say,  ordinary  Euclidean 
geometry  is  about  in  being  about  the  elements  of  the 
space. 

If  we  denote  this  conceptual  space  by  c5»  to  distin- 
guish it  from  (non-geometrizable)  sensible  space  J$i, 
then  the  geometry  of  cSi,  if  constructed  by  means  of 
postulates  P  making  no  indispensable  use  of  algebraic 
analysis,  may  be  called  pure  geometry,  pG*.  If,  as  in 
the  Cartesian  method,  we  use  ordered  number  triads, 
etc.,  as  we  may  use  them,  not  to  be  points,  etc.,  but  to 
represent  points,  etc.,  then  we  get  analytical  geometry, 
aGi,  of  cSi.  On  the  other  hand,  if,  as  we  may,  we  inter- 
pret the  P  by  allowing  the  /-things  to  be  number  triads, 
etc.,  as  above  indicated,  the  resulting  doctrine  is,  not 
geometry,  but  a  pure  algebra  or  analysis,  pA\.  If  we 
use  points,  etc.,  not  to  be,  but  to  represent,  number 
triads,  etc.,  and  so  employ  geometric  language  in  con- 
structing pAi,  we  get  by  this  kind  of  anti-Cartesian 


254    INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

procedure,    not   a   geometry,    but   geometrical   analysis, 

gAz. 

HYPOTHESIS  hi 

It  is  unnecessary  to  say  anything  and  is  not  worth 
while  to  say  much  under  this  hypothesis.  For  if  the 
e's  in  P  do  not  denote  something,  then  as  the  relations 
(if  there  be  any)  are  termless,  the  doctrine  D  (if  there 
be  one)  is  not  about  anything,  unless  about  the  relations, 
but  about  these  it  says  nothing,  for,  if  it  says  aught 
about  them,  what  it  says  is  that  they  are  satisfied  by 
certain  terms  whose  presence  in  the  discourse  is  excluded 
by  hi.  We  may  profitably  say,  however,  that,  in  the 
case  supposed  where  the  e's  do  not  denote  something 
but  are  merely  uninterpreted  variables  ready,  so  to 
speak,  to  denote  something  —  in  this  case  we  may  say 
that,  though  there  is  no  doctrine  D,  there  is  a  doctrinal 
function,  A  (ei,  e2,  ez).  Then  we  should  add  that  the 
doctrines  that  do  arise  from  actualized  possible  interpre- 
tations of  the  e's  are  so  many  values  of  A.  This  func- 
tion A,  if  we  give  some  warning  mark  as  A7  to  its 
symbol,  may  be  further  conveniently  employed  in  talk- 
ing about  an  ambiguous  one  of  the  doctrines  in  question, 
i.  e.,  about  "any  value,"  an  ambiguous  value,  of  the 
function.  As  above  argued,  these  values,  these  doctrines 
are  identical  in  form,  they  are  isomorphic,  all  of  them 
having  the  form  of  A,  but  no  two  of  them  are  the  same 
in  respect  of  content,  reference,  or  meaning.  In  this 
conclusion,  analysis,  happily,  agrees  with  traditional 
usage,  intuition,  and  common  sense. 

CONCLUDING  CONSIDERATIONS 

We  are,  I  believe,  now  prepared  to  answer  definitively 
the  long- vexed  question:  What,  if  any,  sort  of  existence 
have  point  spaces  of  four  or  more  dimensions? 


INTERPRETATIONS  OP  POSTULATE  SYSTEMS    255 

As  we  have  seen,  the  conceptual  space  cS»  of  ordinary 
geometry  is  an  affair  involving  extension;  it  is  a  triply 
extended  conceptual  spread  or  expanse:  three  independ- 
ent linear  extensions  in  it  may  be  chosen;  these  suffice 
to  determine  all  the  others.  So  much  is  as  certain  as 
anything  can  be.  It  is  equally  certain  that  we  can,  for 
we  do  without  meeting  contradiction,  by  means  of  postu- 
lates or  otherwise,  conceive  (not  perceive  or  imagine)  a 
quadruply  extended  spread  or  expanse,  one,  that  is,  in 
which  it  is  possible  to  choose  four  independent  linear 
extensions,  and  then  by  reference  to  these  to  determine 
all  the  rest  There  is  not  the  slightest  difference  in  kind 
among  the  four  independents  and  not  the  slightest  differ- 
ence between  any  three  of  these  and  the  three  of  cSi. 
The  spread  or  expanse  thus  set  up  is  a  c54;  like  cSi,  it 
is  purely  conceptual;  the  extension  it  involves  is,  in  kind, 
identical  with  that  of  cS3',  it  contains  spreads  of  the 
type  of  c.Sa  as  elements  just  exactly  as  a  cS*  contains 
planes  or  spreads  of  type  cSt  as  elements;  it  differs 
not  at  all  from  cS3  except  in  being  one  degree  higher 
in  respect  of  dimensionality.  In  a  word,  cS*  (and,  of 
course,  cS&,  and  so  on)  has  the  same  kind  of  existence  as 
cSz.  It  is  true  that  cSi  is  "imitated"  by  our  sensible 
space  sS),  whilst  there  is  no  sSt  thus  imitating  cSt. 
But  this  writing  is  not  intended  for  one  who  is  capable 
of  thinking  that  the  mentioned  sensible  imitation  or 
instability  of  cSi  confers  upon  the  latter  a  new  or 
peculiar  kind  of  existence. 

But  one  thing  remains  to  be  said,  and  it  is  impor- 
tant. If  one  denies  that  cSa  has  the  conceptually  exten- 
sional  existence,  above  alleged,  then,  of  course,  the 
denial  extends  also  to  cS*,  and  the  two  spaces  are,  in 
respect  of  existence,  still  on  a  level.  If  the  denier  then 
asserts,  and  such  is  the  alternative,  that  cSt  is  only  the 


256     INTERPRETATIONS  OF  POSTULATE  SYSTEMS 

ensemble  of  number  triads,  etc.,  as  above  explained, 
then,  if  he  be  right,  cS*  is  only,  but  equally,  the  ensemble 
of  ordered  quatrains,  etc.,  of  numbers.  Here,  again, 
cSz  and  cSi  have  precisely  the  same  kind  of  existence. 
The  conclusion  is  that  hyperspaces  have  every  kind  of 
existence  that  may  be  warrantably  attributed  to  the  space 
of  ordinary  geometry. 


MATHEMATICAL  PRODUCTIVITY  IN  THE 
UNITED  STATES1 

BOTH  on  its  own  account  and  in  its  relation  to  the 
general  question  of  research,  this  subject  is  naturally 
interesting  to  the  specialists  immediately  concerned; 
and  it  seems  a  happy  augury  that  not  long  ago  several 
western  college  and  university  presidents,  in  convention, 
considered  the  problem  how  to  secure  that  officers  of 
instruction  shall  become,  in  addition,  investigators  and 
producers.  A  complete  solution  will  be  found  when, 
and  only  when,  the  nature  and  importance  of  the  prob- 
lem shall  be  appreciated,  not  only  by  scientific  special- 
ists and  university  presidents,  but  by  educators  and  the 
educated  public  in  general,  and  this  condition  will  be 
satisfied  in  proportion  as  the  interdependence  of  all 
grades  and  varieties  of  educational  and  scientific  activity 
shall  come  to  be  generally  understood,  and  especially 
in  proportion  as  we  learn  to  value  the  things  of  mind, 
not  merely  for  their  utility,  but  for  their  spiritual  worth, 
and  to  seek,  as  a  community,  in  addition  to  comfort 
and  happiness,  the  glory  of  the  sublimer  forms  of  knowl- 
edge and  intellectual  achievement. 

Except  when  the  contrary  may  be  indicated  or  clearly 
implied,  the  discussion  will  confine  itself  to  pure  mathe- 
matics as  distinguished  from  applied  mathematics,  such 
as  mechanics  and  mathematical  physics. 

And  first  as  to  the  significance  of  terms.  According 
to  the  usage  that  has  long  prevailed  among  foreign 

1  Printed  in  the  Educational  Revitv,  November,  190*. 


258  MATHEMATICAL  PRODUCTIVITY 

mathematicians,  and  which,  during  the  last  quarter  of 
a  century,  has  come  to  prevail  also  in  this  country,  the 
term  mathematical  productivity  is  restricted  to  dis- 
covery, successful  research,  extension  in  some  sense  of 
the  boundaries  of  mathematical  knowledge;  and  such 
productive  activity  includes  and  ranges  thru  the  estab- 
lishment of  important  new  theorems,  the  critical  ground- 
ing of  classical  doctrines,  the  discovery  or  invention  of 
new  methods  of  attack,  and,  in  its  highest  form,  the 
opening  and  exploration  of  new  domains. 

Not  only  does  the  term  productivity  now  signify  here 
what  it  signifies  abroad,  but  the  prevailing  standards  in 
the  United  States  agree  with  those  of  Europe.  It  is 
not  meant  that  the  best  work  in  this  country  is  yet 
equal  to  the  very  best  of  the  European,  nor  that  the 
averages  coincide,  but  that  the  Americans  judge  home 
and  foreign  products  by  the  same  canons  of  value,  and 
that  these  are  as  rigorous  as  the  French,  German,  or 
British  rules  of  criticism. 

Time  was  when  productivity  meant,  in  the  United 
States,  the  writing  and  publishing  of  college  text-books  in 
algebra,  geometry,  trigonometry,  analytical  geometry 
and  the  calculus,  not  to  mention  arithmetic.  That  time 
has  gone  by.  At  present  the  term  neither  signifies  such 
work  nor,  except  in  rare  instances,  includes  it.  With 
reverence  for  the  olden  time  when  the  college  professor, 
especially  in  comparison  with  the  average  of  his  suc- 
cessors of  the  present  time,  was  apt  to  be  a  man  of 
general  attainments  and  diversified  learning,  it  may  be 
said  that,  judged  by  modern  standards  of  specialized 
scholarship,  the  special  attainments  of  American  mathe- 
maticians previous  to  a  generation  ago,  except  in  the  case 
of  a  few  illustrious  men,  were  exceedingly  meager  — 
a  fact  which,  as  it  could  hardly  have  been  suspected 


MA  Till-!  MAT  If  A  I.   PRODUCTIVITY  259 

owing  to  their  isolation  by  the  mathematicians  them- 
selves, was  even  less  known  to  their  colleagues  in  other 
branches  of  learning  or  to  the  educated  public  in  general. 
The  writer  of  a  college  text-book  in  mathematics  was 
naturally  regarded  as  a  great  mathematician,  despite 
the  circumstance  that,  in  general,  the  book  contained 
the  sum  of  the  author's  knowledge  of  the  subject 
treated,  much  more  than  the  average  teacher's  knowl- 
edge, and  quite  as  much  as  the  most  capable  youth  was 
expected  to  master  under  the  most  favoring  conditions. 
In  general,  neither  author  nor  teacher  nor  pupil  had 
knowledge  of  the  fact  that  their  most  advanced  instruc- 
tion dealt  only  with  the  rudiments  and  often  even  with 
these  in  an  obsolete  or  obsolescent  manner;  in  general, 
there  was  no  suspicion  that,  on  the  other  side  of  the 
Atlantic,  mathematics  was  a  vast  and  growing  science, 
much  less  that  it  was  developing  so  rapidly  and  in  so 
manifold  a  manner  that  the  greatest  mathematical 
genius  found  it  necessary  to  specialize,  even  in  his  own 
domain.  As  a  natural  consequence  American  mathe- 
matical instruction  depended  almost  exclusively  on  the 
use  of  text-books.  What  was  thus  at  first  a  necessity 
became  a  tradition,  and,  accordingly,  in  striking  con- 
trast with  French  and  German  practice  in  schools  of 
corresponding  grade,  American  college  and  undergraduate 
university  instruction  in  mathematics,  with  some  excep- 
tions, of  which  Harvard  is  the  most  notable,  continues 
still  to  make  the  text-book  the  basis  of  instruction,  even 
where  it  is  not  regarded  as  a  sine  qua  turn  of  the  classroom. 
One  result  of  this  practice  and  tradition  is  that  the  text- 
book, which  early  assumed  in  the  public  estimation  what 
now  seems  to  be  an  exaggerated  importance,  continues 
still  to  be  often  regarded  as  an  indispensable  instrument 
for  the  systematic  impartation  of  knowledge. 


260  MATHEMATICAL  PRODUCTIVITY 

The  text-book  method  in  undergraduate  mathematical 
instruction  undoubtedly  has  some  peculiar  merits  and 
is  recommended  by  considerations  of  weight.  I  am  not 
about  to  advocate  its  abandonment.  That  question, 
moreover^  is  in  a  sense  alien  to  the  subject  here  under 
discussion.  But  I  may  say  in  passing  that  the  notion, 
so  firmly  lodged  in  many  of  our  colleges  and  still  more 
firmly  established  in  the  mind  of  the  general  educated 
public,  that  the  text-book  is  indispensable,  is  an  erro- 
neous one.  That,  as  already  said,  has  been  amply  proved 
both  here  and  abroad,  at  Harvard,  in  some  American 
normal  schools,  and  in  the  schools  of  Germany  and 
France,  by  the  best,  if  not  the  only,  method  available 
for  settling  such  questions,  namely,  by  trial.  And  I 
could  wish  it  were  better  known,  particularly  to  teachers 
in  secondary  schools,  that  some  of  the  ablest  mathe- 
maticians and  teachers  of  mathematics  deprecate,  not 
the  use  of  the  text-book,  for  that  use  has  been  suffi- 
ciently justified  by  the  practice  of  most  eminent  and 
effective  teachers,  but  our  traditional  dependence  upon 
it,  believing  that  this-  dependence  often  hampers  the 
competent  teacher's  freedom  and  so  prevents  a  full 
manifestation  of  the  proper  life  of  the  subject.  For 
the  subject  has  indeed  a  deep  and  serene  and  even  a 
joyous  life,  and,  contrary  to  popular  feeling,  it  is  capable 
of  being  so  interpreted  and  administered  as  to  have, 
not  merely  for  the  few,  but  for  the  many,  for  the 
majority  indeed  of  those  who  find  their  way  to  college, 
not  only  the  highest  disciplinary  value,  which  is  gen- 
erally conceded,  but  a  wonderful  quickening  power  and 
inspiration  as  well.  And  it  may  very  well  be  that  the 
very  great,  tho  not  generally  suspected,  human  signifi- 
cance and  cultural  value  of  mathematics,  the  fact  that 
not  merely  in  its  elements  it  is  highly  useful  and  appli- 


MATHEMATICAL  PRODUCTIVITY  26  X 

cable,  but  that  throughout  the  entire  immensity  and 
wondrous  complex  of  its  development  it  is  informed 
with  beauty,  being  sustained  indeed  by  artistic  interest, 
—  it  may  indeed  be  that  all  this  will  in  some  larger 
measure  come  to  be  felt  and  understood  when  teaching 
shall  depend  less  on  the  text-book,  at  best  a  relatively 
dead  thing,  tending  to  bear  the  spirit  of  instruction 
down,  and  shall  instead  be  more  by  living  men,  speak- 
ing immediately  to  living  men,  out  of  masterful  knowl- 
edge of  their  science  and  with  a  clear  perception  of  its 
spiritual  significance  and  worth. 

To  return  from  this  digression,  it  is  fully  recognized 
by  all  that,  as  undergraduate  mathematical  instruction 
is  now  carried  on  in  our  country,  the  text-book  writer 
is  a  pretty  valuable  citizen,  nor  is  there  any  disposition 
to  detract  from  the  dignity  of  his  activity.  Indeed, 
though  some  of  the  older  books  compare  favorably  in 
important  respects  with  the  best  of  the  new,  it  may  be 
said  that,  in  general,  to  write  a  highly  acceptable  mathe- 
matical book  for  college  use  requires  to-day  an  order 
of  attainment  far  superior  to  that  which  was  sufficient 
even  a  score  or  two  of  years  ago.  The  training  and 
scholarship  which  such  work  presupposes  are,  in  re- 
spect to  amount  and  more  especially  in  respect  to  qual- 
ity, not  only  relatively  great,  but  very  considerable 
absolutely.  The  author  of  the  kind  of  book  in  question 
-and  happily  there  is  no  lack  of  competition  in  this 
field  of  writing  —  may  be  certain  of  intelligent,  if  not 
always  generous,  appreciation;  he  may  be  able  thereby 
to  lengthen  his  purse,  his  book  stands  some  chance  of 
being  briefly  noticed  in  reputable  journals,  and  he  may 
even  gain  local  fame,  but,  however  excellent  the  quality 
of  his  workmanship,  it  will  seldom  secure  him  a  place 
in  the  ranks  of  the  investigator  or  producer.  The  ser- 


262  MATHEMATICAL  PRODUCTIVITY 

vice  of  the  text-book  writer  has  not  been  degraded.  It 
receives  a  more  discriminating  appreciation  than  ever 
before.  It  is  merely  that  this  kind  of  work  has  received 
a  more  critical  appraisement.  Not  a  few  mathematicians 
decline  to  undertake  the  work  of  text-book  writing,  for 
the  reason  that  they  do  not  wish  to  be  classed  as  text- 
book authors.  If  one  who  has  published  several  original 
papers  yields  to  the  temptation  to  write  a  book  for 
college  use,  the  chances  are  his  reputation  will  suffer 
loss  rather  than  gain.  Possibly  such  ought  not  to  be 
the  case,  but  nevertheless  it  is  the  case. 

In  regard  to  mathematical  productivity  proper,  it  is 
probably  true  that  during  the  last  twenty-five  years, 
especially  during  the  latter  half  of  this  period,  there  has 
been  greater  improvement  in  research  work  and  output 
in  this  country  than  elsewhere  in  the  world.  Such 
sweeping  statements  are  of  course  hazardous,  and  I 
make  this  one  subject  to  correction.  At  all  events, 
the  gain  in  question  has  been  great  and  is  full  of  prom- 
ise. Just  about  twenty-five  years  ago  the  American 
Journal  oj  Mathematics  was  founded  at  the  Johns  Hopkins 
University,  where  it  is  still  published  as  a  quarterly. 
Previous  to  that  time  two  or  three  attempts  had  been 
made  to  publish  journals  of  mathematics  in  this  country, 
but  they  met  with  little  success,  and  are  now  scarcely 
remembered.  The  American  Journal  of  Mathematics,  in 
the  beginning,  sought  contributions  from  abroad,  and 
reference  to  the  early  volumes  will  show  that  these  are 
to  a  considerable  extent  occupied  by  foreign  products. 
A  second  journal,  the  Annals  of  Mathematics,  was  founded 
in  1884,  and  published  at  the  University  of  Virginia. 
This  journal,  a  quarterly,  still  flourishes,  being  now 
published  at  Cambridge,  Mass.,  under  the  auspices  of 
Harvard  University.  In  1888  was  founded  the  New 


MATHEMATICAL   PRODUCTIVITY  263 

York  Mathematical  Club,  which  soon  became  the  New 
York  Mathematical  Society  and  began  the  publication 
of  a  monthly  Bulletin.  In  1894  this  society  became  the 
American  Mathematical  Society  which  now  has  a 
membership  of  nearly  four  hundred,  including,  with  few 
exceptions,  every  American  mathematician  of  standing, 
besides  some  members  from  Canada,  England,  and  the 
Continent.  This  society  has  a  rapidly  growing  library, 
and  publishes  two  journals,  the  Bulletin,  already  men- 
tioned, and  the  Transactions,  a  quarterly  journal,  re- 
cently founded,  and  devoted  to  the  publication  of  the 
more  important  results  of  research.  The  four  journals 
named  are,  all  of  them,  of  good  standing  and  exchange 
with  some  of  the  best  British  and  Continental  journals. 
Not  by  any  means  all  the  members  of  the  society  are 
producing  mathematicians,  but  a  large  precentage  of 
them  are  sufficiently  interested  to  attend  one  or  more 
meetings  of  the  society  each  year.  These  meetings  are 
bi-monthly  meetings,  held  in  New  York,  and  a  summer 
meeting  at  a  place  chosen  from  year  to  year.  To  meet 
growing  demands,  a  Chicago  section  has  been  organized, 
which  holds  regular  meetings  in  that  city,  and  a  second 
section  on  the  Pacific  Coast,  whose  business  will  be 
conducted  perhaps  at  San  Francisco.  These  sections 
report  to  the  society  proper,  which  has  its  offices  in 
New  York  City. 

At  these  meetings  there  are  presented  annually  several 
scores  of  papers,  a  percentage  of  which  deal  with  applied 
mathematics.  Of  course  not  all  of  these  papers  are 
important,  but  some  of  them  possess  very  considerable, 
a  few  of  them  distinctly  great,  value,  and  a  large  major- 
ity of  them  fall  properly  within  the  category  of  original 
investigation  as  defined.  In  addition  to  such  more 
regular  contributions,  a  considerable  number  of  mathe- 


264  MATHEMATICAL  PRODUCTIVITY 

matical  papers  are  annually  presented  before  other 
American  scientific  organizations,  as,  for  example,  before 
Section  A  of  the  American  Association  for  the  Advance- 
ment of  Science.  The  majority  of  all  these  articles  are 
found  to  be  available  for  publication,  and  the  result  is 
that,  altho  foreign  contributions  are  no  longer  invited 
as  formerly,  and  few  of  them  received,  the  four  journals 
above  mentioned  are,  nevertheless,  taxed  beyond  their 
capacity;  and,  for  want  of  room,  papers  are  sometimes 
rejected  by  the  American  journals  which,  if  produced 
abroad,  would  probably  be  published  there,  where  the 
facilities  for  publication  are  ampler.  In  fact,  the  number 
of  American  memoirs  published  abroad  exceeds  perhaps 
the  number  of  foreign  contributions  published  here. 

It  is  greatly  to  be  regretted  that  our  facilities  for 
publication,  tho  recently  so  greatly  enhanced,  are  still 
distinctly  inadequate.  For  mathematicians  are  also 
men,  and,  as  such,  one  of  their  most  powerful  incentives 
to  research  is  the  prospect  of  the  recognition  that  comes 
from  having  the  results  of  their  labors  properly  placed 
before  the  scientific  world. 

While  the  picture  "thus  drawn  of  American  mathe- 
matical activity  is  a  pleasing  one  and  is  full  of  encour- 
agement and  hope,  still  we  must  not  disguise  from 
ourselves  the  fact  that,  in  view  of  the  vast  extent  and 
resources  of  our  country  and  of  the  large  number  of 
professional  mathematicians  connected  with  our  numer- 
ous colleges  and  universities,  the  amount  and  the  average 
quality  of  the  American  mathematical  output  are  not 
only  distinctly  inferior  to  that  of  the  more  scientific 
countries  of  Europe,  for  which,  not  without  some  justice 
and  plausibility,  we  are  wont  to  plead  our  youth  in 
defense  and  explanation,  but  this  average  and  amount 
are  by  no  means  a  measure  of  our  native  ability  nor  in 


MATHEMATICAL  PRODUCTIVITY  265 

keeping  with  our  achievements  in  some  other  scarcely 
worthier,  if  less  ethereal,  domains. 

The  reasons  for  this  state  of  case  are  not  far  to  seek, 
and  come  readily  to  light  on  a  minuter  study  of  the 
necessary  and  sufficient  conditions  for  the  vigorous 
prosecution  of  mathematical  research. 

We  may  recall  the  philosopher's  insight  that  "there 
is  but  one  poet  and  that  is  Deity."  The  poet  is  indeed 
born,  we  all  agree;  and  it  is  equally,  if  not  so  obviously, 
true  that  the  great  mathematician  or  financier  or  admin- 
istrator is  born.  But  the  mathematician  is  not  born 
trained  or  born  with  knowledge  of  the  state  of  the 
science,  and  hence  it  goes  without  saying  that  to  native 
ability,  which  we  presuppose  throughout  as  absolutely 
essential  and  which  is  not  so  rare  as  is  often  thought, 
training  must  be  superadded,  years  of  austere  training 
under,  or  still  better,  in  co-operation  with,  competent 
masters  in  a  suitable  atmosphere.  Formerly,  it  was  in 
general  necessary  to  seek  such  training  abroad;  that 
is  no  longer  the  case,  now  that  our  better  universities 
are  manned  with  scholars  of  the  best  American  and 
European  training.  Indeed  the  mathematical  doctorate 
of  a  few  of  our  own  institutions  now  represents  quite 
as  much  as,  if  not  more  than,  the  average  German 
doctorate,  though  less,  we  must  still  confess,  than  the 
French,  which  probably  has  the  highest  significance  of 
any  in  the  world.  Several  of  the  most  highly  productive 
mathematicians  in  the  country  have  not  received  for- 
eign training,  while  a  still  larger  number  of  non-pro- 
ducers studied  abroad  for  years  —  a  fact  showing  that 
such  training  is  neither  a  necessary  nor  a  sufficient 
condition.  It  is  not  intended  to  depreciate  the  absolute 
value  of  foreign  training,  but  only  its  relative  value  — 
its  value  as  compared  with  that  of  the  best  which  our 


266  MATHEMATICAL  PRODUCTIVITY 

own  country  now  affords.  It  is  still  desirable,  when 
not  too  inconvenient,  to  spend  a  year  in  the  atmosphere 
of  foreign  universities,  and  many  avail  themselves  of 
the  opportunity,  largely  for  the  sake  of  the  prestige 
which,  owing  partly  to  a  tradition,  it  still  affords  in 
many  American  communities.  It  is,  of  course,  a  mere 
truism  to  say  that  training,  though  necessary,  is  not  suf- 
ficient. Unless  there  be  the  gift  of  originality,  training 
can  at  best  result  in  receptive  and  critical  scholarship, 
but  not  in  productive  power. 

There  are  in  our  country  a  goodly  number  of  men 
having  the  requisite  ability  and  training,  who,  never- 
theless, produce  but  little  or  nothing  at  all  —  a  fact 
to  be  accounted  for  by  the  absence  in  their  case  of  other 
essential  conditions. 

In  some  cases  library  facilities  are  lacking.  Mathe- 
matical science  is  a  growth.  The  new  rises  out  of  the 
old,  whence  the  necessity  that  the  investigator  have  at 
hand  the  major  part  at  least  of  the  literature  of  his 
subject  from  the  earliest  times.  Even  more  exacting, 
if  possible,  is  the  necessity  of  having  ready  access  to 
the  leading  journals  of  England,  Germany,  France,  and 
Italy,  besides  those  of  America.  One  takes  special 
pleasure  in  mentioning  Italy,  because  she  has  been 
recently  making  rapid  advances  and  in  two  important 
directions,  the  geometry  of  hyperspace  and  mathemat- 
ical logic,  the  ontology  of  pure  thought,  she  comes  well- 
nigh  leading  the  van.  Of  journals  there  are  at  least 
a  dozen  which  are  absolutely  indispensable  to  the  re- 
search worker  and  as  many  more  that  are  highly  desir- 
able. In  addition,  the  producing  mathematician  will 
not  infrequently  have  occasion  to  refer  to  memoirs 
which,  because  of  their  length  or  for  other  reasons,  have 
not  appeared  in  the  journals,  and  are  to  be  found  only 


MATHEMATICAL  PRODUCTIVITY  267 

in  the  proceedings  of  the  leading  general  scientific  and 
philosophical  societies  of  Europe.  The  lack  of  such 
facilities,  which  in  some  cases  a  few  hundred  and  in 
others  a  few  thousand  dollars  would  suffice  to  make  good, 
is  in  itself  sufficient  to  explain  the  non-productivity 
of  not  a  few  American  mathematicians. 

Again,  there  are  cases  where  able  men  have  not  the 
necessary  leisure  to  engage  successfully  in  investiga- 
tion. Our  universities  are  for  the  most  part  so  organ- 
ized that  the  energies  of  scientific  men  are  largely 
expended  in  undergraduate  teaching  and  in  adminis- 
trative work.  We,  as  a  people,  have  yet  to  learn  that 
the  value  of  a  professor  to  a  community  can  be  rightly 
estimated,  not  by  counting  the  number  of  hours  he 
actually  stands  before  his  classes,  but  rather,  if  we  must 
count  at  all,  by  reckoning  the  number  of  hours  devoted 
to  the  preparation  of  his  lectures,  and  more  particularly 
by  the  fruit  of  quiet  study  and  research.  In  Germany 
the  ordinary  professor  lectures  from  four  to  six  hours  a 
week,  to  which  if  we  add  in  some  cases  two  hours 
Seminar iibungcn,  we  have  a  total  of  six  to  eight  hours 
of  presence  in  the  lecture  room.  In  France  the  pro- 
fessor is  expected  to  give  one  course  of  lectures.  These 
take  place  twice  a  week  and  last  from  one  to  one  and  a 
half  hours.  To  this  duty  should  be  added  that  of 
holding  a  large  number  of  examinations  —  a  rather 
wearisome  service  from  which  the  German  escapes. 
When  we  contrast  this  with  the  ten  to  fifteen  and  often 
even  twenty  or  more  hours  of  actual  teaching  demanded 
of  the  American  professor,  to  say  nothing  of  faculty 
meetings,  committee  meetings,  and  the  multitudinous 
examinations,  and  when  we  do  not  fail  to  reflect  that 
ten  hours  are  much  more  than  twice  five  in  their  tax 
upon  energy,  it  is  little  wonder  that  in  productivity  our 


268  MATHEMATICAL   PRODUCTIVITY 

most  brilliant  men  are  often  so  greatly  outclassed  by 
their  foreign  competitors.  Moreover ',  "our  universities 
are  at  present,  for  the  first  two  years,  gymnasia  and 
lycees,  and  our  professors  are  accordingly  obliged  to 
devote  themselves  largely  to  what  is  properly  secondary 
instruction"  —  a  kind  of  work  which,  however  worthy, 
important,  and  necessary,  has  the  effect,  not  merely  of 
drawing  off  the  energy  in  non-productive  channels,  but 
also  eventually  of  forming  and  hardening  the  mind 
about  a  relatively  small  group  of  simpler  ideas. 

Again,  scientific  activity  is  not  infrequently  rendered 
impossible  by  the  amount  of  administrative  work  which 
professors  of  notable  administrative  ability  are  called 
upon  to  perform.  Indeed  "the  problem  presses  for 
solution,  how  to  retain  the  many  peculiar  excellences 
of  our  college  and  university  life  and  at  the  same  time 
to  create  for  certain  men  of  talent  and  training  a  suit- 
able environment  for  the  highest  scientific  activity." 

Once  more,  it  is  very  desirable,  indeed  it  is  really 
necessary,  for  men  working  in  a  branch  of  science  to 
attend  the  meetings  of  scientific  bodies,  in  order  to 
meet  their  fellow-men,  to  take  counsel  of  them,  to  cre- 
ate and  share  in  a  wholesome  esprit  de  corps,  to  catch 
the  inspiration  and  enthusiasm,  and  to  gain  the  sus- 
taining impulses  which  can  come  only  from  personal 
contact  and  co-operation.  But  our  country  is  so  vast, 
the  distances  so  long,  and  traveling  so  expensive,  that 
many  mathematicians,  owing  to  smallness  of  income, 
find  themselves  hopelessly  condemned  to  a  life  of  iso- 
lation, of  which  the  result  is  a  loss  first  of  interest  and 
then  of  power.  It  is  in  vain  that  one  counsels  such 
men  to  wake  up  and  be  strong  and  active,  for  their 
state  of  inactivity  is  less  a  defect  of  will  than  an  effect 
of  circumstances. 


MATHEMATICAL  PRODUCTIVITY  269 

There  is  a  second  phase  of  this  question  of  remunera- 
tion which  is,  happily,  beginning  to  attract  attention 
and  to  receive  consideration  in  university  circles.  I 
refer  to  the  proposition  that  a  wise  economy  will  pro- 
vide, for  university  service,  remuneration,  not  such  as 
would  attract  men  whose  first  ambition  is  to  acquire 
the  ease  that  wealth  is  supposed  to  afford,  but  such  as 
will  not,  by  its  inadequacy  to  the  reasonable  demands 
of  modern  social  life,  deter  men  of  ability  and  predilec- 
tion for  scientific  pursuits  from  entering  upon  them. 
I  know  personally  of  six  young  men,  not  all  of  them 
mathematicians,  who  have  sufficiently  demonstrated 
that  they  possess  such  ability  and  predilection,  five  of 
whom  have  recently  relinquished  the  pursuit  of  science 
and  the  fifth  of  whom  told  me  only  yesterday  that 
he  seriously  contemplates  doing  so,  all  of  them,  for 
the  reason  that,  as  they  allege,  the  university  career 
furnishes  either  not  at  all,  or  too  tardily,  a  financial 
competence  and  consequent  relief  from  practical  condem- 
nation to  celibacy.  It  matters  little  whether  they  be 
mistaken  to  a  degree  or  not,  so  long  as  the  contrary 
conviction  determines  choice.  There  is,  indeed,  more 
than  a  bare  suspicion  that  for  reasons  akin  to  those 
actuating  in  the  cases  cited,  the  university  career, 
particularly  in  case  of  the  more  abstract  sciences,  such 
as  pure  mathematics,  whose  doctrines  have  little  or  no 
market  value,  fails  to  attract  a  due  proportion  of  the 
best  intellects  of  the  country.  For  it  should  be  under- 
stood that  successful  investigation  in  such  sciences 
demands  men  of  intellectual  resource,  of  power,  of  per- 
sistence, in  a  word,  men  of  strenuosity  of  life  and  char- 
acter. Such  men  are  indeed  the  intellectual  peers  of 
the  great  financier,  or  soldier,  or  statesman,  or  admin- 
istrator, and  they  are  aware  of  it;  so  that  if  too  many 


270  MATHEMATICAL  PRODUCTIVITY 

such  men  are  not  to  be  drawn  away  from  scientific  fields 
by  the  prospect  of  achieving  elsewhere  not  only  fame 
but  fortune  also,  it  stands  to  reason  that  the  university 
career  must  promise  at  least  a  competence  and  the 
peace  of  mind  it  brings.  That  such  is  the  case,  and  that 
the  future  will  condemn  the  present  for  a  too  tardy 
recognition  of  the  fact,  is  a  matter  which  can  hardly 
admit  of  doubt. 

That  the  conditions  above  indicated  are  those  which 
determine  the  matter  of  mathematical  productivity  is  a 
proposition  which  not  only  commends  itself  a  priori 
to  the  reason,  but  is  justified  also  a  posteriori  by  experi- 
ence, for  statistics  show  that  those  institutions,  both 
foreign  and  domestic,  where  such  productivity  has 
flourished  best  are  also  those  where  the  conditions  named 
are  most  fully  satisfied,  and  that  where  one,  at  least, 
of  the  conditions  is  not  fulfilled,  there  investigation 
proceeds  but  feebly  or  is  wholly  wanting. 

It  remains  to  mention  another  condition  which  in  a 
sense  includes  all  others,  and  whose  fulfillment  will  come 
gradually  as  at  once  the  cause  and  the  effect  of  the 
fulfillment  of  all  others.  I  mean,  of  course,  a  public 
sentiment  which  will  demand,  because  it  has  learned 
to  appreciate,  knowledge,  not  merely  because  of  its 
applications  and  utility,  but  for  its  beauty,  as  one  appre- 
ciates the  moon  and  the  stars  without  regard  to  their 
aid  in  navigation  —  a  public  sentiment  that  shall  seek 
every  provision  and  regard  as  sacred  every  instrumental- 
ity for  the  advancement  and  ministration  of  knowledge, 
not  only  as  a  means  to  happiness,  but  as  a  glory,  for 
its  own  sake,  as  a  self-justifying  realization  of  the  dis- 
tinctive ambition  of  man,  to  understand  the  universe 
in  which  he  lives  and  the  wondrous  possibilities  of  the 
Reason  unto  which  it  constantly  makes  appeal. 


MATHEMATICS  > 

IN  the  early  part  of  the  last  century  a  philosophic 
French  mathematician,  addressing  himself  to  the  ques- 
tion of  the  perfectibility  of  scientific  doctrines,  expressed 
the  opinion  that  one  may  not  imagine  the  last  word  has 
been  said  of  a  given  theory  so  long  as  it  can  not  by 
a  brief  explanation  be  made  clear  to  the  man  of  the 
street.  Doubtless  that  conception  of  doctrinal  per- 
fectibility, taken  literally,  can  never  be  realized.  For 
doubtless,  just  as  there  exist  now,  so  in  the  future 
there  will  abound,  even  in  greater  and  greater  variety 
and  on  a  vaster  and  vaster  scale,  deep-laid  and  high- 
towering  scientific  doctrines  that,  in  respect  to  their 
infinitude  of  detail  and  in  their  remoter  parts  and  more 
recondite  structure,  shall  not  be  intelligible  to  any  but 
such  as  concentrate  their  life  upon  them.  And  so  the 
noble  dream  of  Gergonne  can  never  literally  come  true. 
Nevertheless,  as  an  ideal,  as  a  goal  of  aspiration,  it 
is  of  the  highest  value,  and,  though  in  no  case  can  it 
be  quite  attained,  it  yet  admits  in  many,  as  I  believe, 
of  a  surprisingly  high  degree  of  approximation.  I  do 

1  An  address  delivered  in  1907  at  Columbia  University,  the  University 
of  Virginia,  Washington  and  Lee  University,  the  University  of  North  Caro- 
lina, Tulane  University,  the  University  of  Arkansas,  the  University  of 
Nebraska,  the  University  of  Missouri,  the  University  of  Chicago,  North- 
western University,  the  University  of  Illinois,  Vanderbilt  University,  the 
University  of  Minnesota,  the  University  of  Michigan,  the  University  of 
Cincinnati,  the  Ohio  State  University,  Vassar  College,  the  University  of 
Vermont,  Purdue  University,  and  the  University  Club  of  New  York  City. 
Printed  by  the  Columbia  University  Press,  1007. 


272  MATHEMATICS 

not  mind  frankly  owning  that  I  do  not  share  in  the 
feeling  of  those,  if  there  be  any  such,  who  regard  their 
special  subjects  as  so  intricate,  mysterious  and  high, 
that  in  all  their  sublimer  parts  they  are  absolutely  in- 
accessible to  the  profane  man  of  merely  general  culture 
even  when  he  is  led  by  the  hand  of  an  expert  and  con- 
descending guide.  For  scientific  theories  are,  each  and 
all  of  them,  and  they  will  continue  to  be,  built  upon  and 
about  notions  which,  however  sublimated,  are  never- 
theless derived  from  common  sense.  These  etherealized 
central  concepts,  together  with  their  manifold  bearings 
on  the  higher  interests  of  life  and  general  thought,  can 
be  measurably  assimilated  to  the  language  of  the  com- 
mon level  from  which  they  arose.  And,  in  passing,  I 
should  like  to  express  the  hope  that  here  at  Columbia 
or  other  competent  center  there  may  one  day  be  estab- 
lished a  magazine  that  shall  have  for  its  aim  to  mediate, 
by  the  help,  if  it  may  be  found,  of  such  pens  as  those 
of  Huxley  and  Clifford,  between  the  focal  concepts  and 
the  larger  aspects  of  the  technical  doctrines  of  the 
specialist,  on  the  one  hand,  and  the  teeming  curiosity, 
the  great  listening,  waiting,  eager,  hungering  con- 
sciousness of  the  educated  thinking  public  on  the  other. 
Such  a  service,  however,  is  not  to  be  lightly  undertaken. 
An  hour,  at  all  events,  is  hardly  time  enough  in  which 
to  conduct  an  excursion  even  of  scientific  folk  through 
the  mazes  of  more  than  twenty  hundred  years  of  mathe- 
matical thought  or  even  to  express  intelligibly,  if  one 
were  competent,  the  significance  of  the  whole  in  a 
critical  estimate. 

Indeed,  such  is  the  character  of  mathematics  in  its 
profounder  depths  and  in  its  higher  and  remoter  zones 
that  it  is  well  nigh  impossible  to  convey  to  one  who  has 
not  devoted  years  to  its  exploration  a  just  impression 


MATHEMATICS  273 

of  the  scope  and  magnitude  of  the  existing  body  of  the 
science.  An  imagination  formed  by  other  disciplines 
and  accustomed  to  the  interests  of  another  field  may 
scarcely  receive  suddenly  an  apocalyptic  vision  of  that 
infinite  interior  world.  But  how  amazing  and  how  edi- 
fying were  such  a  revelation,  if  only  it  could  be  made. 
To  tell  the  story  of  mathematics  from  Pythagoras  and 
Plato  to  Hilbert  and  Lie  and  Poincare";  to  recount  and 
appraise  the  achievements  of  such  as  Euclid  and  Archi- 
medes, Apollonius  and  Diophantus;  to  display  and 
estimate  the  creations  of  Descartes  and  Leibnitz  and 
Newton;  to  dispose  in  genetic  order,  to  analyze,  to 
synthesize  and  evaluate,  the  discoveries  of  the  Ber- 
noullis  and  Euler,  of  Desargues  and  Pascal  and  Monge 
and  Poncelet,  of  Steiner  and  Mobius  and  Pliicker  and 
Staudt,  of  Lobatschewsky  and  Bolyai,  of  W.  R.  Ham- 
ilton and  Grassmann,  of  Laplace,  Lagrange  and  Gauss, 
of  Boole  and  Cayley  and  Hermite  and  Gordan,  of  Bol- 
zano and  Cauchy,  of  Riemann  and  Weierstrass,  of 
Georg  Cantor  and  Boltzmann  and  Klein,  of  the  Peirces 
and  Schroder  and  Peano,  of  Helmholtz  and  Maxwell 
and  Gibbs;  to  explore,  and  then  to  map  for  perspective 
beholding  and  contemplation,  the  continent  of  doctrine 
built  up  by  these  immortals,  to  say  nothing  of  the  count- 
less refinements,  extensions  and  elaborations  meanwhile 
wrought  by  the  genius  and  industry  of  a  thousand 
other  agents  of  the  mathetic  spirit;  —  to  do  that  would 
indeed  be  to  render  an  exceeding  service  to  the  higher 
intelligence  of  the  world,  but  a  service  that  would  re- 
quire the  conjoint  labors  of  a  council  of  scholars  for 
the  space  of  many  years.  Even  the  three  immense 
volumes  of  Moritz  Cantor's  Gesckichte  der  Mathematik, 
though  they  do  not  aspire  to  the  higher  forms  of  elab- 
orate exposition  and  though  they  are  far  from  exhaust- 


274  MATHEMATICS 

ing  the  material  of  the  period  traversed  by  them,  yet 
conduct  the  narrative  down  only  to  I758.1  That  date, 
however,  but  marks  the  time  when  mathematics,  then 
schooled  for  over  a  hundred  eventful  years  in  the  un- 
folding wonders  of  Analytic  Geometry  and  the  Calculus 
and  rejoicing  in  the  possession  of  these  the  two  most 
powerful  among  the  instruments  of  human  thought,  had 
but  fairly  entered  upon  her  modern  career.  And  so 
fruitful  have  been  the  intervening  years,  so  swift  the 
march  along  the  myriad  tracks  of  modern  analysis  and 
geometry,  so  abounding  and  bold  and  fertile  withal 
has  been  the  creative  genius  of  the  time,  that  to  record 
even  briefly  the  discoveries  and  the  creations  since  the 
closing  date  of  Cantor's  work  would  require  an  addition 
to  his  great  volumes  of  a  score  of  volumes  more. 

Indeed  the  modern  developments  of  mathematics 
constitute  not  only  one  of  the  most  impressive,  but  one 
of  the  most  characteristic,  phenomena  of  our  age.  It 
is  a  phenomenon,  however,  of  which  the  boasted  intelli- 
gence of  a  "universalized"  daily  press  seems  strangely 
unaware;  and  there  is  no  other  great  human  interest, 
whether  of  science  or  of  art,  regarding  which  the  mind 
of  the  educated  public  is  permitted  to  hold  so  many 
fallacious  opinions  and  inferior  estimates.  The  golden 
age  of  mathematics  —  that  was  not  the  age  of  Euclid, 
it  is  ours.  Ours  is  the  age  in  which  no  less  than  six 
international  congresses  of  mathematics  have  been  held 
in  the  course  of  nine  years.2  It  is  in  our  day  that  more 
than  a  dozen  mathematical  societies  contain  a  growing 
membership  of  over  two  thousand  men  representing 
the  centers  of  scientific  light  throughout  the  great  cul- 

1  The  work  is  now  being  carried  forward  by  younger  men. 

2  International  congresses  of  mathematicians  are  held  at  intervals  of 
four  years.    Since  the  date  of  this  address  two  have  been  held,  one  at  Rome 
in  1908  and  one  in  Cambridge,  England,  in  1912. 


MATHEMATICS  275 

ture  nations  of  the  world.  It  is  in  our  time  that  over 
five  hundred  scientific  journals  are  each  devoted  in 
part,  while  more  than  two  score  others  are  devoted 
exclusively,  to  the  publication  of  mathematics.  It  is 
in  our  time  that  the  Jahrbuch  iibcr  die  Fortschritte  der 
Mathematik,  though  admitting  only  condensed  abstracts 
with  titles,  and  not  reporting  on  all  the  journals,  has, 
nevertheless,  grown  to  nearly  forty  huge  volumes  in  as 
many  years.  It  is  in  our  time  that  as  many  as  two 
thousand  books  and  memoirs  drop  from  the  mathemat- 
ical press  of  the  world  in  a  single  year,  the  estimated 
number  mounting  up  to  fifty  thousand  in  the  last 
generation.  Finally,  to  adduce  yet  another  evidence  of 
similar  kind,  it  requires  no  less  than  the  seven  ponder- 
ous tomes  of  the  forthcoming  Encyklopadie  der  Mathe- 
matischen  Wissenschaften  to  contain,  not  expositions, 
not  demonstrations,  but  merely  compact  reports  and 
bibliographic  notices  sketching  developments  that  have 
taken  place  since  the  beginning  of  the  nineteenth  cen- 
tury. The  Elements  of  Euclid  is  as  small  a  part  of 
mathematics  as  the  Iliad  is  of  literature;  or  as  the 
sculpture  of  Phidias  is  of  the  world's  total  art.  Indeed 
if  Euclid  or  even  Descartes  were  to  return  to  the  abode 
of  living  men  and  repair  to  a  university  to  resume  pur- 
suit of  his  favorite  study,  it  is  evident  that,  making 
due  allowance  for  his  genius  and  his  fame,  and  pre- 
supposing familiarity  with  the  modern  scientific  lan- 
guages, he  would  yet  be  required  to  devote  at  least  a 
year  to  preparation  before  being  qualified  even  to  begin 
a  single  strictly  graduate  course. 

It  is  not,  however,  by  such  comparisons  nor  by  sta- 
tistical methods  nor  by  any  external  sign  whatever, 
but  only  by  continued  dwelling  within  the  subtle  radi- 
ance of  the  discipline  itself,  that  one  at  length  may 


276  MATHEMATICS 

catch  the  spirit  and  learn  to  estimate  the  abounding 
life  of  modern  mathesis:  oldest  of  the  sciences,  yet 
flourishing  to-day  as  never  before,  not  merely  as  a  giant 
tree  throwing  out  and  aloft  myriad  branching  arms  in 
the  upper  regions  of  clearer  light  and  plunging  deeper 
and  deeper  root  in  the  darker  soil  beneath,  but  rather 
as  an  immense  mighty  forest  of  such  oaks,  which,  how- 
ever, literally  grow  into  each  other  so  that  by  the  junc- 
tion and  intercrescence  of  limb  with  limb  and  root  with 
root  and  trunk  with  trunk  the  manifold  wood  becomes 
a  single  living  organic  growing  whole. 

What  is  this  thing  so  marvelously  vital?  What  does 
it  undertake?  What  is  its  motive?  What  its  signifi- 
cance? How  is  it  related  to  other  modes  and  forms  and 
interests  of  the  human  spirit? 

What  is  mathematics?  I  inquire,  not  about  the  word, 
but  about  the  thing.  Many  have  been  the  answers  of 
former  years,  but  none  has  approved  itself  as  final.  All 
of  them,  by  nature  belonging  to  the  "literature  of 
knowledge,"  have  fallen  under  its  law  and  "perished  by 
supersession."  Naturally  conception  of  the  science 
has  had  to  grow  with  the  growth  of  the  science  itself. 

A  traditional  conception,  still  current  everywhere 
except  in  critical  circles,  has  held  mathematics  to  be  the 
science  of  quantity  or  magnitude,  where  magnitude 
including  multitude  (with  its  correlate  of  number)  as  a 
special  kind,  signified  whatever  was  "capable  of  increase 
and  decrease  and  measurement."  Measurability  was 
the  essential  thing.  That  definition  of  the  science  was 
a  very  natural  one,  for  magnitude  did  appear  to  be  a 
singularly  fundamental  notion,  not  only  inviting  but 
demanding  consideration  at  every  stage  and  turn  of 
life.  The  necessity  of  finding  out  how  many  and  how 
much  was  the  mother  of  counting  and  measurement, 


MATHEMATICS  277 

and  mathematics,  first  from  necessity  and  then  from 
pure  curiosity  and  joy,  so  occupied  itself  with  these 
things  that  they  came  to  seem  its  whole  employment. 

Nevertheless,  numerous  great  events  of  a  hundred 
years  have  been  absolutely  decisive  against  that  view. 
For  one  thing,  the  notion  of  continuum  —  the  "  Grand 
Continuum"  as  Sylvester  called  it  —  that  great  central 
supporting  pillar  of  modern  Analysis,  has  been  con- 
structed by  Weierstrass,  Dedekind,  Georg  Cantor  and 
others,  without  any  reference  whatever  to  quantity,  so 
that  number  and  magnitude  are  not  only  independent, 
they  are  essentially  disparate.  When  we  attempt  to 
correlate  the  two,  the  ordinary  concept  of  measurement 
as  the  repeated  application  of  a  constant  finite  unit, 
undergoes  such  refinement  and  generalization  through 
the  notion  of  Limit  or  its  equivalent  that  counting  no 
longer  avails  and  measurement  retains  scarcely  a  vestige 
of  its  original  meaning.  And  when  we  add  the  further 
consideration  that  non-Euclidean  geometry  employs  a 
scale  in  which  the  unit  of  angle  and  distance,  though  it 
is  a  constant  unit,  nevertheless  appears  from  the  Euclid- 
ean point  of  view  to  suffer  lawful  change  from  step  to 
step  of  its  application,  it  is  seen  that  to  retain  the  old 
words  and  call  mathematics  the  science  of  quantity  or 
magnitude,  and  measurement,  is  quite  inept  as  no 
longer  telling  either  what  the  science  has  actually 
become  or  what  its  spirit  is  bent  upon. 

Moreover,  the  most  striking  measurements,  as  of  the 
volume  of  a  planet,  the  growth  of  cells,  the  valency  of 
atoms,  rates  of  chemical  change,  the  swiftness  of  thought, 
the  penetrative  power  of  radium  emanations,  are  none 
of  them  done  by  direct  repeated  application  of  a  unit  or 
by  any  direct  method  whatever.  They  are  all  of  them 
accomplished  by  one  form  or  another  of  indirection.  It 


278  MATHEMATICS 

was  perception  of  this  fact  that  led  the  famous  phi- 
losopher and  respectable  mathematician,  Auguste  Comte, 
to  define  mathematics  as  "the  science  of  indirect  meas- 
urement." Here  doubtless  we  are  in  presence  of  a 
finer  insight  and  a  larger  view,  but  the  thought  is  not 
yet  either  wide  enough  or  deep  enough.  For  it  is 
obvious  that  there  is  an  immense  deal  of  admittedly 
mathematical  activity  that  is  not  in  the  least  concerned 
with  measurement  whether  direct  or  indirect.  Con- 
sider, for  example,  that  splendid  creation  of  the  nine- 
teenth century  known  as  Projective  Geometry:  a 
boundless  domain  of  countless  fields  where  reals  and 
imaginaries,  finites  and  infinites,  enter  on  equal  terms, 
where  the  spirit  delights  in  the  artistic  balance  and 
symmetric  interplay  of  a  kind  of  conceptual  and  log- 
ical counterpoint,  —  an  enchanted  realm  where  thought 
is  double  and  flows  throughout  in  parallel  streams. 
Here  there  is  no  essential  concern  with  number  or 
quantity  or  magnitude,  and  metric  considerations  are 
entirely  absent  or  completely  subordinate.  The  fact, 
to  take  a  simplest  example,  that  two  points  determine 
a  line  uniquely,  or  that  the  intersection  of  a  sphere  and 
a  plane  is  a  circle,  or  that  any  configuration  whatever  — 
the  reference  is  here  to  ordinary  space  —  presents  two 
reciprocal  aspects  according  as  it  is  viewed  as  an  en- 
semble of  points  or  as  a  manifold  of  planes,  is  not  a 
metric  fact  at  all:  it  is  not  a  fact  about  size  or  quantity 
or  magnitude  of  any  kind.  In  this  domain  it  was 
position  rather  than  size  that  seemed  to  some  the  central 
matter,  and  so  it  was  proposed  to  call  mathematics 
the  science  of  measurement  and  position. 

Even  as  thus  expanded,  the  conception  yet  excludes 
many  a  mathematical  realm  of  vast  extent.  Consider 
that  immense  class  of  things  known  as  Operations. 


MATHEMATICS  279 

These  are  limitless  alike  in  number  and  in  kind.  Now 
it  so  happens  that  there  are  many  systems  of  operations 
such  that  any  two  operations  of  a  given  system,  if 
thought  as  following  one  another,  together  thus  produce 
the  same  effect  as  some  other  single  operation  of  the 
system.  Such  systems  are  infinitely  numerous  and 
present  themselves  on  every  hand.  For  a  simple  illus- 
tration, think  of  the  totality  of  possible  straight  motions 
in  space.  The  operation  of  going  from  point  A  to 
point  B,  followed  by  the  operation  of  going  from  B 
to  point  C,  is  equivalent  to  the  single  operation  of  going 
straight  from  A  to  C.  Thus  the  system  of  such  opera- 
tions is  a  closed  system:  combination,  i.  e.,  of  any  two 
of  the  operations  yields  a  third  one,  not  without,  but 
within,  the  system.  The  great  notion  of  Group,  thus 
simply  exemplified,  though  it  had  barely  emerged  into 
consciousness  a  hundred  years  ago,  has  meanwhile 
become  a  concept  of  fundamental  importance  and  pro- 
digious fertility,  not  only  affording  the  basis  of  an 
imposing  doctrine  —  the  Theory  of  Groups  —  but  there- 
with serving  also  as  a  bond  of  union,  a  kind  of  connec- 
tive tissue,  or  rather  as  an  immense  cerebro-spinal 
system,  uniting  together  a  large  number  of  widely  dis- 
similar doctrines  as  organs  of  a  single  body.  But  - 
and  this  is  the  point  to  be  noted  here  —  the  abstract 
operations  of  a  group,  though  they  are  very  real  things, 
are  neither  magnitudes  nor  positions. 

This  way  of  trying  to  come  to  an  adequate  conception 
of  mathematics,  namely,  by  attempting  to  characterize 
in  succession  its  distinct  domains,  or  its  varieties  of 
content,  or  its  modes  of  activity,  in  the  hope  of  finding 
a  common  definitive  mark,  is  not  likely  to  prove  suc- 
cessful. For  it  demands  an  exhaustive  enumeration, 
not  only  of  the  fields  now  occupied  by  the  science,  but 


280  MATHEMATICS 

also  of  those  destined  to  be  conquered  by  it  in  the  future, 
and  such  an  achievement  would  require  a  prevision  that 
none  may  claim. 

Fortunately  there  are  other  paths  of  approach  that 
seem  more  promising.  Everyone  has  observed  that 
mathematics,  whatever  it  may  be,  possesses  a  certain 
mark,  namely,  a  degree  of  certainty  not  found  elsewhere. 
So  it  is,  proverbially,  the  exact  science  par  excellence. 
Exact,  no  doubt,  but  in  what  sense?  An  excellent 
answer  is  found  in  a  definition  given  about  one  genera- 
tion ago  by  a  distinguished  American  mathematician, 
Professor  Benjamin  Peirce:  "Mathematics  is  the  science 
which  draws  necessary  conclusions "  —  a  formulation  of 
like  significance  with  the  following  fine  mot  by  Professor 
William  Benjamin  Smith:  "Mathematics  is  the  uni- 
versal art  apodictic."  These  statements,  though  neither 
of  them  is  adequate,  are  both  of  them  telling  approxima- 
tions, at  once  foreshadowing  and  neatly  summarizing 
for  popular  use,  the  epoch-making  thesis  established  by 
the  creators  of  modern  logic,  namely,  that  mathematics 
is  included  in,  and,  in  a  profound  sense,  may  be  said 
to  be  identical  with,  Symbolic  Logic.  Observe  that  the 
emphasis  falls  on  the  quality  of  being  "necessary," 
i.  e.,  correct  logically,  or  valid  formally. 

But  why  are  mathematical  conclusions  correct?  Is 
it  that  the  mathematician  has  a  reasoning  faculty  essen- 
tially different  in  kind  from  that  of  other  men?  By 
no  means.  What,  then,  is  the  secret?  Reflect  that  con- 
clusion implies  premises,  that  premises  involve  terms, 
that  terms  stand  for  ideas  or  concepts  or  notions,  and 
that  these  latter  are  the  ultimate  material  with  which 
the  spiritual  architect,  called  the  Reason,  designs  and 
builds.  Here,  then,  one  may  expect  to  find  light.  The 
apodictic  quality  of  mathematical  thought,  the  correct- 


MATHEMATICS  281 

ness  of  its  conclusions  as  conclusions,  are  due,  not  to 
any  special  mode  of  ratiocination,  but  to  the  character 
of  the  concepts  with  which  it  deals.  What  is  that  dis- 
tinctive characteristic?  The  answer  is:  precision  and 
completeness  of  determination.  But  how  comes  the 
mathematician  by  such  completeness?  There  is  no 
mysterious  trick  involved:  some  concepts  admit  of 
such  precision  and  completeness,  others  do  not;  the 
mathematician  is  one  who  deals  with  those  that  do. 

The  matter,  however,  is  not  quite  so  simple  as  it 
sounds,  and  I  bespeak  your  attention  to  a  word  of 
caution  and  of  further  explanation.  The  ancient  maxim, 
ex  nihilo  nikil  fit,  may  well  be  doubted  where  it  seems 
most  obviously  valid,  namely,  in  the  realm  of  matter, 
for  it  may  be  that  matter  has  evolved  from  something 
else;  but  the  maxim  cannot  be  ultimately  denied  where 
its  application  is  least  obvious,  namely,  in  the  realm  of 
mind,  for  without  principia  in  the  strictest  sense,  doc- 
trine is,  in  the  strictest  sense,  impossible.  And  when 
the  mathematician  speaks  of  complete  determination  of 
concepts  and  of  rigor  of  demonstration,  he  does  not 
mean  that  the  undefined  and  the  undemonstrated  have 
been  or  can  be  entirely  eliminated  from  the  foundations 
of  his  science.  He  knows  that  such  elimination  is  im- 
possible; he  knows,  too,  that  it  is  unnecessary',  for  some 
undefinable  ideas  are  perfectly  clear  and  some  undemon- 
strable  propositions  are  perfectly  precise  and  certain. 
It  is  in  terms  of  such  concepts  that  a  definable  notion, 
if  it  is  to  be  mathematically  available,  must  admit  of 
complete  determination,  and  in  terms  of  such  proposi- 
tions that  mathematical  discourse  secures  its  rigor.  It 
is,  then,  of  such  indefinables  among  ideas  and  such  in- 
demonstrable* among  propositions  —  paradoxical  as  the 
statement  may  appear  —  that  the  foundations  of  mathe- 


282  MATHEMATICS 

matics  in  its  ideal  conception  are  composed;  and  what- 
ever doctrine  is  logically  constructive  on  such  a  basis 
is  mathematics  either  actually  or  potentially.  I  am  not 
asserting  that  the  substructure  herewith  characterized 
has  been  brought  to  completion.  It  is  on  the  concep- 
tion of  it  that  the  accent  is  here  designed  to  fall,  for 
it  is  the  conception  as  such  that  at  once  affords  to 
fundamental  investigation  a  goal  and  a  guide  and  fur- 
nishes the  means  of  giving  the  science  an  adequate 
definition. 

On  the  other  hand,  actually  to  realize  the  conception 
requires  that  the  foundation  to  be  established  shall  both 
include  every  element  that  is  essential  and  exclude  every 
one  that  is  not.  For  a  foundation  that  subsequently 
demands  or  allows  superfoetation  of  hypotheses  is  in- 
complete; and  one  that  contains  the  non-essential  is 
imperfect.  Of  the  two  problems  thus  presented,  it  is 
the  latter,  the  problem  of  exclusion,  of  reducing  prin- 
ciples to  a  minimum,  of  applying  Occam's  Razor  to  the 
pruning  away  of  non-essentials,  —  it  is  that  problem 
that  taxes  most  severely  both  the  analytic  and  the  con- 
structive powers  of  criticism.  And  it  is  to  the  solution 
of  that  problem  that  the  same  critical  spirit  of  our  time, 
which  in  other  fields  is  reconstructing  theology,  burning 
out  the  dross  from  philosophy,  and  working  relentless 
transformations  of  thought  on  every  hand,  has  directed 
a  chief  movement  of  modern  mathematics. 

Apart  from  its  technical  importance,  which  can 
scarcely  be  overestimated,  the  power,  depth  and  com- 
prehensiveness of  the  modern  critical  movement  in 
mathematics,  make  it  one  of  the  most  significant  sci- 
entific phenomena  of  the  last  century.  Double  in 
respect  to  origin,  the  movement  itself  has  been  com- 
posite. One  component  began  at  the  very  center  of 


MATHEMATICS  283 

mathematical  activity,  while  the  other  took  its  rise  in 
what  was  then  erroneously  regarded  as  an  alien  do- 
main, the  great  domain  of  symbolic  logic. 

A  word  as  to  the  former  component.  For  more  than 
a  hundred  years  after  the  inventions  of  Analytical 
Geometry  and  the  Calculus,  mathematicians  may  be 
said  to  have  fairly  rioted  in  applications  of  these  instru- 
ments to  physical,  mechanical  and  geometric  problems, 
without  concerning  themselves  about  the  nicer  questions 
of  fundamental  principles,  cogency,  and  precision.  In 
the  latter  part  of  the  eighteenth  century  the  efforts  of 
Euler,  Lacroix  and  others  to  systematize  results  served 
to  reveal  in  a  startling  way  the  necessity  of  improving 
foundations.  Constructive  work  was  not  indeed  arrested 
by  that  disclosure.  On  the  contrary  new  doctrines 
continued  to  rise  and  old  ones  to  expand  and  flourish. 
But  a  new  spirit  had  begun  to  manifest  itself.  The 
science  became  increasingly  critical  as  its  towering  edi- 
fices more  and  more  challenged  attention  to  their  foun- 
dations. Manifest  already  in  the  work  of  Gauss  and 
Lagrange,  the  new  tendence,  under  the  powerful  impulse 
and  leadership  of  Cauchy,  rapidly  develops  into  a 
momentous  movement.  The  Calculus,  while  its  instru- 
mental efficacy  is  meanwhile  marvelously  improved,  is 
itself  advanced  from  the  level  of  a  tool  to  the  rank  and 
dignity  of  a  science.  The  doctrines  of  the  real  and  of 
the  complex  variable  are  grounded  with  infinite  patience 
and  care,  so  that,  owing  chiefly  to  the  critical  construc- 
tive genius  of  Weierstrass  and  his  school,  that  stateliest 
of  all  the  pure  creations  of  the  human  intellect  —  the 
Modern  Theory  of  Functions  with  its  manifold  branches 
-  rests  to-day  on  a  basis  not  less  certain  and  not  less 
enduring  than  the  very  integers  with  which  we  count. 
The  movement  still  sweeps  on,  not  only  extending  to 


284  MATHEMATICS 

all  the  cardinal  divisions  of  Analysis  but,  through  the 
agencies  of  such  as  Lobatschewsky  and  Bolyai,  Grass- 
mann  and  Riemann,  Cayley  and  Klein,  Hilbert  and  Lie, 
recasting  the  foundations  of  Geometry  also.  And  there 
can  scarcely  be  a  doubt  that  the  great  domains  of 
Mechanics  and  Mathematical  Physics  are  by  their 
need  destined  to  a  like  invasion. 

In  the  light  of  all  this  criticism,  mathematics  came  to 
appear  as  a  great  ensemble  of  theories,  compendent 
no  doubt,  interpenetrating  each  other  in  a  wondrous 
way,  yet  all  of  them  distinct,  each  built  up  by  logical 
processes  on  its  own  appropriate  basis  of  pure  hy- 
potheses, or  assumptions,  or  postulates.  As  all  the 
theories  were  thus  seen  to  rest  equally  on  hypothetical 
foundations,  all  were  seen  to  be  equally  legitimate;  and 
doctrines  like  those  of  Quaternions,  non-Euclidean 
geometry  and  Hyperspace,  for  a  time  suspected  because 
based  on  postulates  not  all  of  them  traditional,  speedily 
overcame  their  heretical  reputations  and  were  admitted 
to  the  circle  of  the  lawful  and  orthodox. 

It  is  one  thing,  however,  to  deal  with  the  principal 
divisions  of  mathematics  severally,  underpinning  each 
with  a  foundation  of  its  own.  That,  broadly  speaking, 
has  been  the  plan  and  the  effect  of  the  critical  move- 
ment as  thus  far  sketched.  But  it  is  a  very  different 
and  a  profounder  thing  to  underlay  all  the  divisions  at 
once  with  a  single  foundation,  with  a  foundation  that 
shall  serve  as  a  support,  not  merely  for  all  the  divisions 
but  for  something  else,  distinct  from  each  and  from  the 
sum  of  all,  namely,  for  the  whole,  the  science  itself, 
which  they  constitute.  It  is  nothing  less  than  that 
achievement  which,  unconsciously  at  first,  consciously 
at  last,  has  been  the  aim  and  goal  of  the  other  compo- 
nent of  the  critical  movement,  that  component  which, 


MATHEMATICS  285 

as  already  said,  found  its  origin  and  its  initial  interest 
in  the  field  of  symbolic  logic.  The  advantage  of  em- 
ploying symbols  in  the  investigation  and  exposition 
of  the  formal  laws  of  thought  is  not  a  recent  discovery. 
As  everyone  knows,  symbols  were  thus  employed  to  a 
small  extent  by  the  Stagirite  himself.  The  advantage, 
however,  was  not  pursued;  because  for  two  thousand 
years  the  eyes  of  logicians  were  blinded  by  the  blazing 
genius  of  the  "master  of  those  that  know."  With  the 
single  exception  of  the  reign  of  Euclid,  the  annals  of 
science  afford  no  match  for  the  tyranny  that  has  been 
exercised  by  the  logic  of  Aristotle.  Even  the  important 
logical  researches  of  Leibnitz  and  Lambert  and  their 
daring  use  of  symbolical  methods  were  powerless  to 
break  the  spell.  It  was  not  till  1854  when  George 
Boole,  having  invented  an  algebra  to  trace  and  illumi- 
nate the  subtle  ways  of  reason,  published  his  symbolical 
"Investigation  of  the  Laws  of  Thought,"  that  the  revo- 
lution in  logic  really  began.  For,  although  for  a  time 
neglected  by  logicians  and  mathematicians  alike,  it 
was  Boole's  work  that  inspired  and  inaugurated  the 
scientific  movement  now  known  and  honored  throughout 
the  world  under  the  name  of  Symbolic  Logic. 

It  is  true,  the  revolution  has  advanced  in  silence. 
The  discoveries  and  creations  of  Boole's  successors,  of 
C.  S.  Peirce,  of  Schroeder,  of  Peano  and  of  their  dis- 
ciples and  peers,  have  not  been  proclaimed  by  the  daily 
press.  Commerce  and  politics,  gossip  and  sport,  acci- 
dent and  crime,  the  shallow  and  transitory  affairs  of 
the  exoteric  world, —  these  have  filled  the  columns  and 
left  no  room  to  publish  abroad  the  deep  and  abiding 
things  achieved  in  the  silence  of  cloistral  thought.  The 
demonstration  by  symbolical  means  of  the  fact  that  the 
three  laws  of  Identity,  Excluded  Middle  and  Non- 


286  MATHEMATICS 

contradiction  are  absolutely  independent,  none  of  them 
being  derivable  from  the  other  two;  the  discovery  that 
the  syllogism  is  not  deducible  from  those  laws  but  has 
to  be  postulated  as  an  independent  principle;  the  dis- 
covery of  the  astounding  and  significant  fact  that  false 
propositions  imply  all  propositions  and  that  true  ones, 
though  not  implying,  are  implied  by,  all;  the  discovery 
that  most  reasoning  is  not  syllogistic,  but  is  asyllogistic, 
in  form,  and  that,  therefore,  contrary  to  the  teaching 
of  tradition,  the  class-logic  of  Aristotle  is  not  adequate 
to  all  the  concerns  of  rigorous  thought;  the  discovery 
that  Relations,  no  less  than  Classes,  demand  a  logic  of 
their  own,  and  that  a  similar  claim  is  valid  in  the  case 
of  Propositions:  no  intelligence  of  these  events  nor  of 
the  immense  multitude  of  others  which  they  but  mea- 
gerly  serve  to  hint  and  to  exemplify,  has  been  cabled 
round  the  world  and  spread  broadcast  by  the  flying 
bulletins  of  news.  Even  the  scientific  public,  for  the 
most  part  accustomed  to  viewing  the  mind  as  only  the 
instrument  and  not  as  a  subject  of  study,  has  been 
slow  to  recognize  the  achievements  of  modern  research 
in  the  minute  anatomy  of  thought.  Indeed  it  has 
been  not  uncommon  for  students  of  natural  science  to 
sneer  at  logic  as  a  stale  and  profitless  pursuit,  as  the 
barren  mistress  of  scholastic  minds.  These  men  have 
not  been  aware  of  what  certainly  is  a  most  profound, 
if  indeed  it  be  not  the  most  significant,  scientific  move- 
ment of  our  time.  In  America,  in  England,  in  Germany, 
in  France,  and  especially  in  Italy  —  supreme  histolo- 
gist  of  the  human  understanding  —  the  deeps  of  mind 
and  logical  reality  have  been  explored  in  our  genera- 
tion as  never  before  in  the  history  of  the  world.  Owing 
to  the  power  of  the  symbolic  method,  not  only  the  founda- 
tions of  the  Aristotelian  logic  —  the  Calculus  of  Classes  — 


MATHEMATICS  287 

have  been  recast,  but  side  by  side  with  that  everlasting 
monument  of  Greek  genius,  there  rise  today  other  struc- 
tures, fit  companions  of  the  ancient  edifice,  namely,  the 
Logic  of  Relations  and  the  Logic  of  Propositions. 

And  what  are  the  entities  that  have  been  found  to 
constitute  the  base  of  that  triune  organon?  The  answer 
is  surprising:  a  score  or  so  of  primitive,  indemonstrable, 
propositions  together  with  less  than  a  dozen  undefinable 
notions,  called  logical  constants.  But  what  is  more 
surprising  —  for  here  we  touch  the  goal  and  are  enabled 
to  enunciate  what  has  been  justly  called  "one  of  the 
greatest  discoveries  of  our  age"  -is  the  fact  that  the 
basis  of  logic  is  the  basis  of  mathematics  also.  Thus 
the  two  great  components  of  the  critical  movement, 
though  distinct  in  origin  and  following  separate  paths, 
are  found  to  converge  at  last  in  the  thesis:  Symbolic 
Logic  is  Mathematics,  Mathematics  is  Symbolic  Logic, 
the  twain  are  one. 

Is  it  really  so?  Does  the  identity  exist  in  fact?  Is 
it  true  that  so  simple  a  unifying  foundation  for  what  has 
hitherto  been  supposed  two  distinct  and  even  mutually 
alien  interests  has  been  actually  ascertained?  The 
basal  masonry  is  indeed  not  yet  completed  but  the  work 
has  advanced  so  far  that  the  thesis  stated  is  beyond 
dispute  or  reasonable  doubt.  Primitive  propositions 
appear  to  allow  some  freedom  of  choice,  questions  still 
exist  regarding  relative  fundamentally,  and  statements 
of  principles  have  not  yet  crystallized  into  settled  and 
final  form;  but  regarding  the  nature  of  the  data  to  be 
assumed,  the  smallness  of  their  number  and  their  ade- 
quacy, agreement  is  substantial.  In  England,  Russell 
and  Whitehead  f  are  successfully  engaged  now  in  forging 

1  This  work  has  been  projected  in  four  immense  volumes  bearing  the 
title,  Prindpia  Matkematua,  of  which  three  volumes  have  appeared. 


288  MATHEMATICS 

"chains  of  deduction"  binding  the  cardinal  matters 
of  Analysis  and  Geometry  to  the  premises  of  General 
Logic,  while  in  Italy  the  Formulaire  de  Mathematiques 
of  Peano  and  his  school  has  been  for  some  years  grow- 
ing into  a  veritable  encyclopedia  of  mathematics  wrought 
by  the  means  and  clad  in  the  garb  of  symbolic  logic. 

But  is  it  not  incredible  that  the  concept  of  number 
with  all  its  distinctions  of  cardinal  and  ordinal,  frac- 
tional and  whole,  rational  and  irrational,  algebraic  and 
transcendental,  real  and  complex,  finite  and  infinite, 
and  the  concept  of  geometric  space,  in  all  its  varieties 
of  form  and  dimensionality,  is  it  not  incredible  that 
mathematical  ideas,  surpassing  in  multitude  the  sands 
of  the  sea,  should  be  precisely  definable,  each  and  all 
of  them,  in  terms  of  a  few  logical  constants,  in  terms, 
i.  e.,  of  such  indefinable  notions  as  such  that,  implica- 
tion, denoting,  relation,  class,  prepositional  function,  and 
two  or  three  others?  And  is  it  not  incredible  that  by 
means  of  so  few  as  a  score  of  premises  (composed  of 
ten  principles  of  deduction  and  ten  other  indemonstrable 
propositions  of  a  general  logical  nature),  the  entire 
body  of  mathematical  doctrine  can  be  strictly  and  for- 
mally deduced? 

It  is  wonderful,  indeed,  but  not  incredible.  Not  in- 
credible in  a  world  where  the  mustard  seed  becometh  a 
tree,  not  incredible  in  a  world  where  all  the  tints  and 
hues  of  sea  and  land  and  sky  are  derived  from  three 
primary  colors,  where  the  harmonies  and  the  melodies 
of  music  proceed  from  notes  that  are  all  of  them  but 
so  many  specifications  of  four  generic  marks,  and  where 
three  concepts  —  energy,  mass,  motion,  or  mass,  time, 
space  —  apparently  suffice  for  grasping  together  in 
organic  unity  the  mechanical  phenomena  of  a  universe. 

But  the  thesis  granted,  does  it  not  but  serve  to  justify 


MATHEMATICS  289 

the  cardinal  contentions  of  the  depredators  of  mathe- 
matics? Does  it  not  follow  from  it  that  the  science 
is  only  a  logical  grind,  suited  only  to  narrow  and  strait- 
ened intellects  content  to  tramp  in  treadmill  fashion 
the  weary  rounds  of  deduction?  Does  it  not  follow  that 
Schopenhauer  was  right  in  regarding  mathematics  as 
the  lowest  form  of  mental  activity,  and  that  he  and  our 
own  genial  and  enlightened  countryman,  Oliver  Wendell 
Holmes,  were  right  in  likening  mathematical  thought 
to  the  operations  of  a  calculating  machine?  Does  it 
not  follow  that  Huxley's  characterization  of  mathematics 
as  "that  study  which  knows  nothing  of  observation, 
nothing  of  induction,  nothing  of  experiment,  nothing 
of  causation,"  is  surprisingly  confirmed  by  fact?  Does 
it  not  follow  that  Sir  William  Hamilton's  famous  and 
terrific  diatribe  against  the  science  finds  ample  warrant 
in  truth?  Does  it  not  follow,  as  the  Scotch  philosopher 
maintains,  that  mathematics  regarded  as  a  discipline, 
as  a  builder  of  mind,  is  inferior?  That  devotion  to  it 
is  fatal  to  the  development  of  the  sensibilities  and  the 
imagination?  That  continued  pursuit  of  the  study 
leaves  the  mind  narrow  and  dry,  meagre  and  lean, 
disqualifying  it  both  for  practical  affairs  and  for  those 
large  and  liberal  studies  where  moral  questions  inter- 
vene and  judgment  depends,  not  on  nice  calculation  by 
rule,  but  on  a  wide  survey  and  a  balancing  of 
probabilities? 

The  answer  is,  No.  Those  things  not  only  do  not 
follow  but  they  are  not  true.  Every  count  in  the  in- 
dictment, whether  explicit  or  only  implied,  is  false. 
Not  only  that,  but  the  opposite  in  each  case  is  true. 
On  that  point  there  can  be  no  doubt;  authority,  reason 
and  fact,  history  and  theory,  are  here  in  perfect  accord. 
Let  me  say  once  for  all  that  I  am  conscious  of  no  desire 


2QO  MATHEMATICS 

to  exaggerate  the  virtues  of  mathematics.  I  am  willing 
to  admit  that  mathematicians  do  constitute  an  important 
part  of  the  salt  of  the  earth.  But  the  science  is  no 
catholicon  for  mental  disease.  There  is  in  it  no  power 
for  transforming  mediocrity  into  genius.  It  cannot 
enrich  where  nature  has  impoverished.  It  makes  no 
pretense  of  creating  faculty  where  none  exists,  of  open- 
ing springs  in  desert  minds.  "Du  bist  am  Ende  —  was 
du  bist."  The  great  mathematician,  like  the  great 
poet  or  great  naturalist  or  great  administrator,  is  born. 
My  contention  shall  be  that  where  the  mathetic  endow- 
ment is  found,  there  will  usually  be  found  associated 
with  it,  as  essential  implications  in  it,  other  endowments 
in  generous  measure,  and  that  the  appeal  of  the  science 
is  to  the  whole  mind,  direct  no  doubt  to  the  central 
powers  of  thought,  but  indirectly  through  sympathy 
of  all,  rousing,  enlarging,  developing,  emancipating  all, 
so  that  the  faculties  of  will,  of  intellect  and  feeling, 
learn  to  respond,  each  in  its  appropriate  order  and 
degree,  like  the  parts  of  an  orchestra  to  the  "urge  and 
ardor"  of  its  leader  and  lord. 

As  for  Hamilton  and  Schopenhauer,  those  detractors 
need  not  detain  us  long.  Indeed  but  for  their  fame  and 
the  great  influence  their  opinions  have  exercised  over 
"the  ignorant  mass  of  educated  men,"  they  ought  not 
in  this  connection  to  be  noticed  at  all.  Of  the  subject 
on  which  they  presumed  to  pronounce  authoritative 
judgment  of  condemnation,  they  were  both  of  them 
ignorant,  the  former  well  nigh  proudly  so,  the  latter 
unawares,  but  both  of  them,  in  view  of  their  pretensions, 
disgracefully  ignorant.  Lack  of  knowledge,  however,  is 
but  a  venial  sin,  and  English-speaking  mathematicians 
have  been  disposed  to  hope  that  Hamilton  might  be 
saved  in  accordance  with  the  good  old  catholic  doc- 


MATHEMATICS  2QI 

trine  of  invincible  ignorance.  But  even  that  hope,  as 
we  shall  see,  must  be  relinquished.  In  1853  William 
Whewell,  then  fellow  and  tutor  of  Trinity  College, 
Cambridge,  published  an  appreciative  pamphlet  entitled 
"Thoughts  on  the  Study  of  Mathematics  as  a  Part  of 
a  Liberal  Education."  The  author  was  a  brilliant 
scholar.  "Science  was  his  forte,"  but  "omniscience  his 
foible,"  and  his  reputation  for  universal  knowledge  was 
looming  large.  That  reputation,  however,  Hamilton  re- 
garded as  his  own  prerogative.  None  might  dispute  the 
claim,  much  less  share  the  glory  of  having  it  acknowl- 
edged on  his  own  behalf.  Whewell  must  be  crushed. 
In  the  following  year  Sir  William  replies  in  the  Edin- 
burg  Review,  and  such  a  show  of  learning!  The  reader 
is  apparently  confronted  with  the  assembled  opinions 
of  the  learned  world,  and  —  what  is  more  amazing  - 
they  all  agree.  Literati  of  every  kind,  of  all  nations 
and  every  tongue,  orators,  philosophers,  educators, 
scientific  men,  ancient  and  modern,  known  and  unknown, 
all  are  made  to  support  Hamilton's  claim,  and  even  the 
most  celebrated  mathematicians  seem  eager  to  declare 
that  the  study  of  mathematics  is  unworthy  of  genius 
and  injures  the  mind.  Whewell  was  overwhelmed, 
reduced  to  silence.  His  promised  rejoinder  failed  to 
appear.  The  Scotchman's  victory  was  complete,  his 
fame  enhanced,  and  his  alleged  judgment  regarding  a 
great  human  interest  of  which  he  was  ignorant  has 
reigned  over  the  minds  of  thousands  of  men  who  have 
been  either  willing  or  constrained  to  depend  on  borrowed 
estimates.  But  even  all  this  may  be  condoned.  Jeal- 
ousy, vanity,  parade  of  learning,  may  be  pardoned  even 
in  a  philosopher.  Hamilton's  deadly  sin  was  none  of 
these,  it  was  sinning  against  the  light.  In  October, 
1877,  A.  T.  Bledsoe,  then  editor  of  the  Southern  Renew 


2  Q2  MATHEMATICS 

—  unfortunately  too  little  known  —  published  an  article 
in  that  journal  in  which  he  proved  beyond  a  reasonable 
doubt  —  I  have  been  at  the  pains  to  verify  the  proof  — 
that  Hamilton  by  studied  selections  and  omissions  de- 
liberately and  maliciously  misrepresented  the  great 
authors  from  whom  he  quoted  —  d'Alembert,  Blaise 
Pascal,  Descartes  and  others  —  distorting  their  express 
and  unmistakable  meaning  even  to  the  extent  of  com- 
plete inversion.  This  same  verdict  regarding  Hamilton's 
vandalism,  in  so  far  as  it  relates  to  the  works  of 
Descartes,  was  independently  reached  by  Professor 
Pringsheim  and  in  1904  announced  by  him  in  his 
Festrede  before  the  Munich  Academy  of  Sciences.  As 
for  Schopenhauer,  I  regret  to  say  that  a  similar  charge 
and  finding  stand  against  him  also.  For  not  only  did 
he  endorse  without  examination  and  re-utter  Hamilton's 
tirade  in  the  strongest  terms,  thus  reinforcing  it  and 
giving  it  currency  on  the  continent,  but,  as  Pringsheim 
has  shown,  the  German  philosopher,  by  careful  excision 
from  the  writings  of  Lichtenberg,  converts  that 
distinguished  physicist's  just  strictures  on  the  then  flour- 
ishing but  wayward  Combinatorial  School  of  mathe- 
matics into  a  severe  condemnation  of  mathematicians 
in  general  and  of  the  science  itself,  which,  nevertheless, 
in  the  opening  but  omitted  line  of  the  very  passage 
from  which  Schopenhauer  quotes,  is  characterized  by 
Lichtenberg  as  "eine  gar  herrliche  Wissenschaft."  Re- 
garding the  question  of  the  intrinsic  merit  of  the  esti- 
mate of  mathematics  which  these  two  most  famous  and 
influential  enemies  of  the  science  have  made  so  largely 
current  in  the  world  that  it  fairly  fills  the  atmosphere 
and  people  take  it  in  unconsciously  as  by  a  kind  of  cere- 
bral suction,  I  shall  speak  in  another  connection.  What 
I  desire  to  emphasize  here  is  the  fact  that  neither  the 


MATHEMATICS  2  93 

vast,  splendid,  superficial  learning  of  the  pompous 
author  of  "The  Philosophy  of  the  Conditioned"  nor 
the  pungence  and  pith,  brilliance  and  intrepidity  of 
the  author  of  "Die  Welt  als  Wille"  can  avail  to  con- 
stitute either  of  them  an  authority  in  a  subject  in  which 
neither  was  informed  and  in  which  both  stand  convicted 
falsifiers  of  the  judgments  and  opinions  of  other  men. 

As  to  Huxley  and  Holmes,  the  case  is  different.  Both 
of  them  were  generous,  genial  and  honest,  and  to  their 
opinions  on  any  subject  we  gladly  pay  respect  qualified 
only  as  the  former's  judgment  regarding  mathematics 
was  qualified  by  Sylvester  himself: 

"VersULndige  Leute  kannst  du  irren  schn 
In  Sachen  n&mlich,  die  sic  nicht  verstehn." 

In  relation  to  Huxley's  statement  that  mathematical 
study  knows  nothing  of  observation,  induction,  exper- 
iment, and  causation,  it  ought  to  be  borne  in  mind 
that  there  are  two  kinds  of  observation:  outer  and 
inner,  objective  and  subjective,  material  and  immaterial, 
sensuous  and  sense- transcending;  observation,  that  is, 
of  physical  things  by  the  bodily  senses,  and  observation, 
by  the  inner  eye,  by  the  subtle  touch  of  the  intellect, 
of  the  entities  that  dwell  in  the  domain  of  logic  and 
constitute  the  objects  of  pure  thought.  For,  phrase  it 
as  you  will,  there  is  a  world  that  is  peopled  with  ideas, 
ensembles,  propositions,  relations,  and  implications, 
in  endless  variety  and  multiplicity,  in  structure  ranging 
from  the  very  simple  to  the  endlessly  intricate  and 
complicate.  That  world  is  not  the  product  but  the  object, 
not  the  creature  but  the  quarry  of  thought,  the  entities 
composing  it  —  propositions,  for  example,  —  being  no 
more  identical  with  thinking  them  than  wine  is  identical 
with  the  drinking  of  it.  Mind  or  no  mind,  that  world 


294  MATHEMATICS 

exists  as  an  extra-personal  affair,  —  Pragmatism  to  the 
contrary  notwithstanding.  It  appears  to  me  to  be  a 
radical  error  of  pragmatism  to  blink  the  fact  that  the 
most  fundamental  of  spiritual  things,  namely,  curiosity, 
never  poses  as  a  maker  of  truth  but  is  found  always  and 
only  in  the  attitude  of  seeking  it.  Indeed  truth  might 
be  denned  to  be  the  presupposition  or  the  complement 
of  curiosity  —  as  that  without  which  curiosity  would 
cease  to  be  what  it  is.  The  constitution  of  that  extra- 
personal  world,  its  intimate  ontological  make-up,  is 
logic  in  its  essential  character  and  substance  as  an  inde- 
pendent and  extra-personal  form  of  being,  while  the 
study  of  that  constitution  is  logic  pragmatically,  in  its 
character,  i.  e.,  as  an  enterprise  of  mind.  Now  —  and 
this  is  the  point  I  wish  to  stress  —  just  as  the  astron- 
omer, the  physicist,  the  geologist,  or  other  student  of 
objective  science  looks  abroad  in  the  world  of  sense, 
so,  not  metaphorically  speaking  but  literally,  the  mind 
of  the  mathematician  goes  forth  into  the  universe  of 
logic  in  quest  of  the  things  that  are  there;  exploring 
the  heights  and  depths  for  facts  —  ideas,  classes,  rela- 
tionships, implications,  and  the  rest;  observing  the 
minute  and  elusive  with'  the  powerful  microscope  of  his 
Infinitesimal  Analysis;  observing  the  elusive  and  vast 
with  the  limitless  telescope  of  his  Calculus  of  the  In- 
finite; making  guesses  regarding  the  order  and  internal 
harmony  of  the  data  observed  and  collocated;  testing 
the  hypotheses,  not  merely  by  the  complete  induction 
peculiar  to  mathematics,  but,  like  his  colleagues  of  the 
outer  world,  resorting  also  to  experimental  tests  and 
incomplete  induction;  frequently  finding  it  necessary, 
in  view  of  unforeseen  disclosures,  to  abandon  a  once 
hopeful  hypothesis  or  to  transform  it  by  retrenchment 
or  by  enlargement  : —  thus,  in  his  own  domain,  matching, 


MATHEMATICS  295 

point  for  point,  the  processes,  methods  and  experience 
familiar  to  the  devotee  of  natural  science. 

Is  it  replied  that  it  was  not  observation  of  the  objects 
of  pure  thought  but  the  other  kind,  namely,  sensuous 
observation,  that  Huxley  had  in  mind,  then  I  rejoin 
that,  nevertheless,  observation  by  the  inner  eye  of  the 
things  of  thought  is  observation,  not  less  genuine,  not 
less  difficult,  not  less  rich  in  its  objects  and  disciplinary 
value,  than  is  sensuous  observation  of  the  things  of 
sense.  But  this  is  not  all,  nor  nearly  all.  Indeed  for 
direct  beholding,  for  immediate  discerning,  of  the  things 
of  mathematics  there  is  none  other  light  but  one,  namely, 
psychic  illumination,  but  mediately  and  indirectly  they 
are  often  revealed  or  at  all  events  hinted  by  their  sensu- 
ous counterparts,  by  indications  within  the  radiance  of 
day,  and  it  is  a  great  mistake  to  suppose  that  the 
mathetic  spirit  elects  as  its  agents  those  who,  having 
eyes,  yet  see  not  the  things  that  disclose  themselves  in 
solar  light.  To  facilitate  eyeless  observation  of  his 
sense- transcending  world,  the  mathematician  invokes 
the  aid  of  physical  diagrams  and  physical  symbols  in 
endless  variety  and  combination;  the  logos  is  thus 
drawn  into  a  kind  of  diagrammatic  and  symbolical  in- 
carnation, gets  itself  externalized,  made  flesh,  so  to 
speak;  and  it  is  by  attentive  physical  observation  of 
this  embodiment,  by  scrutinizing  the  physical  frame  and 
make-up  of  his  diagrams,  equations  and  formulae,  by 
experimental  substitutions  in,  and  transformations  of, 
them,  by  noting  what  emerges  as  essential  and  what  as 
accidental,  the  things  that  vanish  and  those  that  do  not, 
the  things  that  vary  and  the  things  that  abide  un- 
changed, as  the  transformations  proceed  and  trains 
of  algebraic  evolution  unfold  themselves  to  view,  —  it 
is  thus,  by  the  laboratory  method,  by  trial  and  by 


296  MATHEMATICS 

watching,  that  often  the  mathematician  gains  his  best 
insight  into  the  constitution  of  the  invisible  world  thus 
depicted  by  visible  symbols.  Indeed  the  importance  to 
the  mathematician  of  such  sensuous  observation  cannot 
be  overrated.  It  is  not  merely  that  the  craving  to  see 
has  led  to  the  construction  of  the  manifold  models, 
ingenious  and  noble,  of  Schilling  and  others,  illustrating 
important  parts  of  Higher  Geometry,  Analysis  Situs, 
Function  Theory  and  other  doctrines,  but  the  annals 
of  the  science  are  illustrious  with  achievements  made 
possible  by  facts  first  noted  by  the  physical  eye.  To 
take  a  simple  example  from  ancient  days,  it  was  by 
observation  of  the  fact  that  the  squares  of  certain 
numbers  are  each  the  sum  of  two  other  squares,  the 
detection  and  collection  of  these  numbers  by  the  method 
of  trial,  observation  of  the  fact  that  apparently  all  and 
only  the  numbers  of  such  triplets  are  measures  of  the 
sides  of  right  triangles,  —  it  was  thus,  by  observation 
and  experiment,  by  the  method  of  incomplete  induction, 
common  to  the  experimental  sciences,  that  the  Pyth- 
agorean theorem,  now  familiar  throughout  the  world, 
was  discovered.  It  was  by  Leibnitz's  observation  of 
the  definitely  lawful  .manner  in  which  the  coefficients  of 
a  system  of  equations  enter  their  solution  that  the 
suggestion  came  of  a  notion  on  the  basis  of  which 
there  has  grown  up  in  our  time  an  imposing  theory, 
an  algebra  built  up  on  algebra  —  the  colossal  doctrine 
of  Determinants.  It  was  the  observation,  the  detection 
by  the  eye  of  Lagrange  and  Boole  and  Eisenstein,  of  the 
fact  that  linear  transformation  of  certain  algebraic 
expressions  leaves  certain  functions  of  their  coefficients 
absolutely  undisturbed  in  form,  unaltered  in  frame  of 
constitution,  that  gave  rise  to  the  concept,  and  there- 
with to  the  morphological  doctrine,  of  Invariants,  a 


MATHEMATICS  2Q7 

theory  filling  the  heavens  like  a  light-bearing  ether, 
penetrating  all  the  branches  of  geometry  and  analysis, 
revealing  everywhere  abiding  configurations  in  the  midst 
of  change,  everywhere  disclosing  the  eternal  reign  of  the 
law  of  Form.  It  was  in  order  to  render  evident  to 
sensuous  observation  and  to  keep  constantly  before  the 
physical  eye  the  pervasive  symmetry  of  mathematical 
thought  that  Hesse  in  the  employment  of  homogeneous 
coordinates  set  the  example,  since  then  generally  fol- 
lowed, of  replacing  a  variety  of  different  letters  by 
repetitions  of  a  single  one  distinguished  by  indices  or 
subscripts,  —  a  practice  yet  further  justified  on  grounds 
both  of  physical  and  of  intellectual  economy.  It  was 
by  sensuous  observation  that  Clerk  Maxwell,  in  the 
beginning  of  his  wondrous  career,  detected  a  lack  of 
symmetry  in  the  then  recognized  equations  of  electro- 
dynamics and  by  that  observed  fact  together  with  a 
discriminating  sense  of  the  scientific  significance  of 
esthetic  intimations,  that  he  was  led  to  remove  the 
seeming  blemish  by  the  addition  of  a  term,  antedating 
experimental  justification  of  his  daring  deed  by  twenty 
years:  an  example  of  prescience  not  surpassed  by  that 
of  Adams  and  Leverrier  who,  while  engaged  in  the  study 
of  planetary  disturbance,  each  of  them  about  the  same 
time  and  independently  of  the  other,  felt  the  then  un- 
known Neptune  "trembling  on  the  delicate  thread  of 
their  analysis"  and  correctly  informed  the  astronomer 
where  to  point  his  telescope  in  order  to  behold  the 
planet.  One  might  go  on  to  cite  the  theorem  of  Sturm 
in  Equation  Theory,  the  "Diophantine  theorems  of 
Fermat"  in  the  Theory  of  Numbers,  the  Jacobian  "doc- 
trine of  double  periodicity"  in  Function  Theory,  Le- 
gendre's  law  of  reciprocity,  Sylvester's  reduction  of 
Eider's  problem  of  the  Virgins  to  the  form  of  a  question 


298  MATHEMATICS 

in  Simple  Partitions,  and  so  on  and  on,  thus  continuing 
indefinitely  the  story  of  the  great  r61e  of  observation, 
experiment  and  incomplete  induction,  in  mathematical 
discovery.  Indeed  it  is  no  wonder  that  even  Gauss, 
"facile  princeps  matematicorum,"  even  though  he  dwelt 
aloft  in  the  privacy  of  a  genius  above  the  needs  and 
ways  of  other  minds,  yet  pronounced  mathematics  "a 
science  of  the  eye." 

Indeed  the  time  is  at  hand  when  at  least  the  academic 
mind  should  discharge  its  traditional  fallacies  regarding 
the  nature  of  mathematics  and  thus  in  a  measure  pro- 
mote the  emancipation  of  criticism  from  inherited 
delusions  respecting  the  kind  of  activity  in  which  the 
life  of  the  science  consists.  Mathematics  is  no  more 
the  art  of  reckoning  and  computation  than  architecture 
is  the  art  of  making  bricks  or  hewing  wood,  no 
more  than  painting  is  the  art  of  mixing  colors  on  a 
palette,  no  more  than  the  science  of  geology  is  the  art 
of  breaking  rocks,  or  the  science  of  anatomy  the  art  of 
butchering. 

Did  not  Babbage  or  somebody  invent  an  adding 
machine?  And  does  it  not  follow,  say  Holmes  and 
Schopenhauer,  that  mathematical  thought  is  a  merely 
mechanical  process?  Strange  how  such  trash  is  occa- 
sionally found  in  the  critical  offering  of  thoughtful  men 
and  thus  acquires  circulation  as  golden  coin  of  wisdom. 
It  would  not  be  sillier  to  argue  that,  because  Stanley 
Jevons  constructed  a  machine  for  producing  certain  forms 
of  logical  inference,  therefore  all  thought,  even  that  of 
a  philosopher  like  Schopenhauer  or  that  of  a  poet  like 
Holmes,  is  merely  a  thing  of  pulleys  and  levers  and 
screws,  or  that  the  pianola  serves  to  prove  that  a  sym- 
phony by  Beethoven  or  a  drama  by  Wagner  is  reducible 
to  a  trick  of  mechanics. 


MATHEMATICS  299 

But  far  more  pernicious,  because  more  deeply  im- 
bedded and  persistent,  is  the  fallacy  that  the  mathe- 
matician's mind  is  but  a  syllogistic  mill  and  that  his 
life  resolves  itself  into  a  weary  repetition  of  A  is  B,  B 
is  C,  therefore  A  is  C;  and  Q.E.D.  That  fallacy  is  the 
Carthago  delenda  of  regnant  methodology.  Reasoning, 
indeed,  in  the  sense  of  compounding  propositions  into 
formal  arguments,  is  of  great  importance  at  every  stage 
and  turn,  as  in  the  deduction  of  consequences,  in  the 
testing  of  hypotheses,  in  the  detection  of  error,  in  pur- 
ging out  the  dross  from  crude  material,  in  chastening 
the  deliverances  of  intuition,  and  especially  in  the  final 
stages  of  a  growing  doctrine,  in  welding  together  and 
concatenating  the  various  parts  into  a  compact  and  co- 
herent whole.  But,  indispensable  in  all  such  ways  as 
syllogistic  undoubtedly  is,  it  is  of  minor  importance  and 
minor  difficulty  compared  with  the  supreme  matters 
of  Invention  and  Construction.  Begrijfbildung,  the 
resolution  of  the  nebula  of  consciousness  into  star-forms 
of  definite  ideas;  discriminating  sensibility  to  the  log- 
ical significances,  affinities  and  bearings  of  these;  sus- 
ceptibility to  the  delicate  intimations  of  the  subtle  or  the 
remote;  sensitiveness  to  dim  and  fading  tremors  sent 
below  by  breezes  striking  the  higher  sails;  the  ability 
to  grasp  together  and  to  hold  in  steady  view  at  once  a 
multitude  of  ideas,  to  transcend  the  individuals  and, 
compounding  their  forces,  to  seize  the  resultant  mean- 
ing of  them  all ;  the  ability  to  summon  not  only  concepts 
but  doctrines,  marshalling  them  and  bringing  them  to 
bear  upon  a  single  point,  like  great  armies  converging 
to  a  critical  center  on  a  battle  field.  These  and  such 
as  these  are  the  powers  that  mathematical  activity  in 
its  higher  rdles  demands.  The  power  of  ratiocination, 
as  already  said,  is  of  exceeding  great  importance  but 


300  MATHEMATICS 

it  is  neither  the  base  nor  the  crown  of  the  faculties 
essential  to  "  Mathematicised  Man."  When  the  greatest 
of  American  logicians,  speaking  of  the  powers  that  con- 
stitute the  born  geometrician,  had  named  Conception, 
Imagination,  and  Generalization,  he  paused.  There- 
upon from  one  in  the  audience  there  came  the  challenge, 
"What  of  Reason?"  The  instant  response,  not  less 
just  than  brilliant,  was  "Ratiocination  —  that  is  but 
the  smooth  pavement  on  which  the  chariot  rolls." 
When  the  late  Sophus  Lie,  great  comparative  anatomist 
of  geometric  theories,  creator  of  the  doctrines  of  Contact 
Transformations,  and  Infinite  Continuous  Groups,  and 
revolutionizer  of  the  Theory  of  Differential  Equations, 
was  asked  to  name  the  characteristic  endowment  of  the 
mathematician,  his  answer  was  the  following  quaternion: 
Phantasie,  Energie,  Selbstvertrauen,  Selbstkritik.  Not  a 
word,  you  observe,  about  ratiocination.  Phantasie,  not 
merely  the  fine  frenzied  fancy  that  gives  to  airy  nothings 
a  local  habitation  and  a  name,  but  the  creative  imagina- 
tion that  conceives  ordered  realms  and  lawful  worlds 
in  which  our  own  universe  is  as  but  a  point  of  light 
in  a  shining  sky;  Energie,  not  merely  endurance  and 
doggedness,  not  persistence  merely,  but  mental  vis  viva, 
the  kinetic,  plunging,  penetrating  power  of  intellect; 
Selbstvertrauen  and  Selbstkritik,  self-confidence  aware  of 
its  ground,  deepened  by  achievement  and  reinforced 
until  in  men  like  Richard  Dedekind,  Bernhard  Bolzano 
and  especially  Georg  Cantor  it  attains  to  a  spiritual  bold- 
ness that  even  dares  leap  from  the  island  shore  of  the 
Finite  over  into  the  all-surrounding  boundless  ocean  of 
Infinitude  itself,  and  thence  brings  back  the  gladdening 
news  that  the  shoreless  vast  of  Transfinite  Being  differs 
in  its  logical  structure  from  that  of  our  island  home  only 
in  owning  the  reign  of  more  generic  law. 


MATHEMATICS  301 

Indeed  it  is  not  surprising,  in  view  of  the  polydynamic 
constitution  of  the  genuinely  mathematical  mind,  that 
many  of  the  major  heroes  of  the  science,  men  like 
Desargues  and  Pascal,  Descartes  and  Leibnitz,  Newton, 
Gauss,  and  Bolzano,  Helmholtz  and  Clifford,  Riemann 
and  Salmon  and  Plucker  and  Poincarl,  have  attained 
to  high  distinction  in  other  fields  not  only  of  science 
but  of  philosophy  and  letters  too.  And  when  we  reflect 
that  the  very  greatest  mathematical  achievements  have 
been  due,  not  alone  to  the  peering,  microscopic,  histo- 
logic  vision  of  men  like  Weierstrass,  illuminating  the 
hidden  recesses,  the  minute  and  intimate  structure  of 
logical  reality,  but  to  the  larger  vision  also  of  men  like 
Klein  who  survey  the  kingdoms  of  geometry  and  analysis 
for  the  endless  variety  of  things  that  flourish  there,  as 
the  eye  of  Darwin  ranged  over  the  flora  and  fauna  of 
the  world,  or  as  a  commercial  monarch  contemplates 
its  industry,  or  as  a  statesman  beholds  an  empire;  when 
we  reflect  not  only  that  the  Calculus  of  Probability  is  a 
creation  of  mathematics  but  that  the  master  mathe- 
matician is  constantly  required  to  exercise  judgment  - 
judgment,  that  is,  in  matters  not  admitting  of  cer- 
tainty —  balancing  probabilities  not  yet  reduced  nor 
even  reducible  perhaps  to  calculation;  when  we  reflect 
that  he  is  called  upon  to  exercise  a  function  analogous 
to  that  of  the  comparative  anatomist  like  Cuvier,  com- 
paring theories  and  doctrines  of  every  degree  of  similar- 
ity and  dissimilarity  of  structure;  when,  finally,  we 
reflect  that  he  seldom  deals  with  a  single  idea  at  a  time, 
but  is  for  the  most  part  engaged  in  wielding  organized 
hosts  of  them,  as  a  general  wields  at  once  the  divisions 
of  an  army  or  as  a  great  civil  administrator  directs  from 
his  central  office  diverse  and  scattered  but  related  groups 
of  interests  and  operations;  then,  I  say,  the  current 


302  MATHEMATICS 

opinion  that  devotion  to  mathematics  unfits  the  devotee 
for  practical  affairs  should  be  known  for  false  on  a 
priori  grounds.  And  one  should  be  thus  prepared  to 
find  that  as  a  fact  Gaspard  Monge,  creator  of  descrip- 
tive geometry,  author  of  the  classic  "Applications  de 
Panalyse  a  la  geometric";  Lazare  Carnot,  author  of  the 
celebrated  works,  "Geometrie  de  position,"  and  "Re- 
flexions sur  la  Metaphysique  du  Calcul  infinitesimal"; 
Fourier,  immortal  creator  of  the  "Theorie  analytique 
de  la  chaleur";  Arago,  rightful  inheritor  of  Monge's 
chair  of  geometry;  and  Poncelet,  creator  of  pure  pro- 
jective  geometry;  one  should  not  be  surprised,  I  say, 
to  find  that  these  and  other  mathematicians  in  a  land 
sagacious  enough  to  invoke  their  aid,  rendered,  alike 
in  peace  and  in  war,  eminent  public  service. 

To  speak  at  length,  if  that  were  necessary,  of  Huxley's 
deliverance  that  the  study  of  mathematics  "knows 
nothing  of  causation,"  the  "law  of  my  song  and  the 
hastening  hour  forbid."  Suffice  it  to  say  in  passing 
that  when  the  mathematician  seeks  the  consequences 
of  given  suppositions,  saying  'when  these  precede, 
those  will  follow/  and  when,  having  plied  a  circle,  a 
sphere  or  other  form  chosen  from  among  infinitudes  of 
configurations,  with  some  transformation  among  infinite 
hosts  at  his  disposal,  he  speaks  of  its  'effect/  then,  I 
submit,  he  is  employing  the  language  of  causation 
with  as  nice  propriety  as  it  admits  of  in  a  world  where, 
as  everyone  knows,  except  such  as  still  enjoy  the  bless- 
ings of  a  juvenile  philosophy,  the  best  we  can  say  is 
that  the  ceaseless  shuttles  fly  back  and  forth,  and 
streams  of  events  without  original  source  flow  on  with- 
out ultimate  termination.  Indeed  it  is  a  certain  and 
signal  lesson  of  science  in  all  its  forms  everywhere  that 
the  language  of  cause  and  effect,  except  in  the  sense  of 


MATHEMATICS  303 

facts   being   lawfully   implied   in   other   facts,    has   no 
indispensable  use. 

I  have  not  spoken  of  "Applied  Mathematics,"  and 
that  for  the  best  of  reasons:  there  is,  strictly  speaking, 
no  such  thing.  The  term  indeed  exists,  and,  in  a  con- 
servative practical  world  that  cares  but  little  for  "The 
nice  sharp  quillets  of  the  law,"  it  will  doubtless  persist 
as  a  convenient  designation  for  something  that  never 
existed  and  never  can.  It  is  of  the  very  essence  of  the 
practician  type  of  mind  not  to  know  aught  as  it  is  in 
itself  nor  aught  as  self-justified  but  to  mistake  the 
secondary  and  accidental  for  the  primary  and  essential, 
to  blink  and  elude  the  presence  of  immediate  worth, 
and  being  thus  blind  to  instant  and  immanent  ends, 
to  revel  in  means  and  uses  and  applications,  requiring 
all  things  to  excuse  their  being  by  extraneous  and 
emanant  effects,  —  vindicating  the  stately  elm  by  its 
promise  of  lumber,  or  the  lily  by  its  message  of  purity, 
or  the  flood  of  Niagara  by  its  available  energy,  or  even 
knowledge  itself  by  the  worldly  advantage  and  the  power 
which  it  gives.  I  am  told  that  even  the  deep  and  ex- 
quisite terminology  of  art  has  been  to  some  extent 
invaded  by  such  barbarous  and  shallow  phrases  as 
'applied  music,'  'applied  architecture,'  'applied  sculp- 
ture,' 'applied  painting/  as  if  Beauty,  virgin  mother  of 
art,  could,  without  dissolution  of  her  essential  char- 
acter, consciously  become  the  willing  drudge  and  para- 
mour of  Use.  And  I  suppose  we  are  fated  yet  to  hear 
of  applied  glory,  applied  holiness,  applied  poetry  - 
t.  e.,  poetry  that  is  consciously  pedagogic  or  that  aims 
at  a  moral  and  thereby  sinks  or  rises  to  the  level  of  a 
sermon  —  of  applied  joy,  applied  ontology,  yea,  of 
applied  inapplicability  itself. 

It   is   in   implications   and   not   in   applications   that 


304  MATHEMATICS 

mathematics  has  its  lair.  Applied  mathematics  is  mathe- 
matics simply  or  is  not  mathematics  at  all.  To  think 
aright  is  no  characteristic  striving  of  a  class  of  men;  it 
is  a  common  aspiration;  and  Mechanics,  Mathematical 
Physics,  Mathematical  Astronomy,  and  the  other  chief 
Anwendungsgebiete  of  mathematics,  as  Geodesy,  Geo- 
physics, and  Engineering  in  its  various  branches,  are  all 
of  them  but  so  many  witnesses  of  the  truth  of  Riemann's 
saying  that  "Natural  science  is  the  attempt  to  com- 
prehend nature  by  means  of  exact  concepts."  A  gas 
molecule  regarded  as  a  minute  sphere  or  other  geometric 
form,  however  complicate;  stars  and  planets  conceived  as 
ellipsoids  or  as  points,  and  their  orbits  as  loci;  time  and 
space,  mass  and  motion  and  impenetrability;  velocity, 
acceleration  and  energy;  the  concepts  of  norm  and 
average;  —  what  are  these  but  mathematical  notions? 
And  the  wondrous  garment  woven  of  them  in  the  loom 
of  logic  —  what  is  that  but  mathematics?  Indeed 
every  branch  of  so-called  applied  mathematics  is  a 
mixed  doctrine,  being  thoroughly  analyzable  into  two 
disparate  parts:  one  of  these  consists  of  determinate 
concepts  formally  combined  in  accordance  with  the 
canons  of  logic,  i.  e.,  it  is  mathematics  and  not  natural 
science  viewed  as  matter  of  observation  and  experiment; 
the  other  is  such  matter  and  is  natural  science  in  that 
conception  of  it  and  not  mathematics.  No  fibre  of 
either  component  is  a  filament  of  the  other.  It  is  a 
fundamental  error  to  regard  the  term  Mathematicisa- 
tion  of  thought  as  the  importation  of  a  tool  into  a 
foreign  workshop.  It  does  not  signify  the  transition  of 
mathematics  conceived  as  a  thing  accomplished  over  into 
some  outlying  domain  like  physics,  for  example.  Its 
significance  is  different  radically,  far  deeper  and  far 
wider.  It  means  the  growth  of  mathematics  itself,  its 


MATHEMATICS  305 

extension  and  development  from  within;  it  signifies 
the  continuous  revelation,  the  endlessly  progressive 
coming  into  view,  of  the  static  universe  of  logic;  or, 
to  put  it  dynamically,  it  means  the  evolution  of  intel- 
lect, the  upward  striving  and  aspiration  of  thought 
everywhere,  to  the  level  of  cogency,  precision  and  exacti- 
tude. This  self-propagation  of  the  rational  logos,  the 
springing  up  of  mathetic  rigor  even  in  void  and  formless 
places,  in  the  very  retreats  of  chaos,  is  to  my  mind  the 
most  impressive  and  significant  phenomenon  in  the 
history  of  science,  and  never  so  strikingly  manifest  as 
in  the  last  half  hundred  years.  Seventy-two  years  ago, 
even  Comte,  the  stout  advocate  of  mathematics  as 
constituting  "the  veritable  point  of  departure  for  all 
rational  scientific  education,  general  or  special,"  ex- 
pressed the  opinion  that  we  should  never  "be  in  posi- 
tion by  any  means  whatever  to  study  the  chemical 
composition  of  the  stars."  In  less  than  twenty-five 
years  thereafter  that  negative  prophecy  was  falsified 
by  the  chemical  genius  of  Bunsen  fortified  by  the  mathe- 
matics of  Kirchoff.  Not  only  has  mathematics  grown, 
in  the  domain  of  Physics,  into  the  vast  proportions  of 
Rational  Dynamics,  but  the  derivative  and  integral  of 
the  Calculus,  and  Differential  Equations,  are  more  and 
more  finding  subsistence  in  Chemistry  also,  and  by  the 
work  of  Nernst  and  others  even  the  foundations  of  the 
latter  science  are  being  laid  in  mathematico-physical 
considerations.  Merely  to  sketch  most  briefly  the 
mathematical  literature  that  has  grown  up  in  the  field 
of  Political  Economy  requires  twenty-five  pages  of  the 
above  mentioned  Encyklop&dit  of  mathematics.  Similar 
sketches  for  Statistics  and  Life  Insurance  require  no 
less  than  thirty  and  sixty-five  pages  respectively.  Even 
in  the  baffling  and  elusive  matter  of  Psychology,  the 


306  MATHEMATICS 

work  of  Herbart,  Fechner,  Weber,  Wundt  and  others 
confirms  the  hope  that  the  soil  of  that  great  field  will 
some  day  support  a  vigorous  growth  of  mathematics. 
It  seems  indeed  as  if  the  entire  surface  of  the  world  of 
human  consciousness  were  predestined  to  be  covered 
over,  in  varying  degrees  of  luxuriance,  by  the  flora  of 
mathetic  science. 

But  while  mathematics  may  spring  up  and  flourish 
in  any  and  all  experimental  and  observational  fields,  it 
is  by  no  means  to  be  expected  that  'experiment  and 
observation'  will  ever  thus  be  superseded.  Such  domains 
are  rather  destined  to  be  occupied  at  the  same  time  by 
two  tenants,  mathematical  science  and  science  that  is 
not  mathematical.  But  while  the  former  will  serve  as 
an  ideal  standard  for  the  latter,  mathematics  has 
neither  the  power  nor  the  disposition  to  disseize  experi- 
ment and  observation  of  any  holdings  that  are  theirs 
by  the  rights  of  conquest  and  use.  Between  mathe- 
matics on  the  one  hand  and  non-mathematical  science 
on  the  other,  there  can  never  occur  collision  or  quarrel, 
for  the  reason  that  the  two  interests  are  ultimately 
discriminated  by  the  kind  of  curiosity  whence  they 
spring.  The  mathematician  is  curious  about  definite 
naked  relationships,  about  logically  possible  modes  of 
order,  about  varieties  of  implication,  about  completely 
determined  or  determinable  functional  relationships, 
considered  solely  in  and  of  themselves,  considered,  that 
is,  without  the  slightest  concern  about  any  question 
whether  or  no  they  have  any  external  or  sensuous 
validity  or  other  sort  of  validity  than  that  of  being 
logically  thinkable.  It  is  the  aggregate  of  things  think- 
able logically  that  constitutes  the  mathematician's 
universe,  and  it  is  inconceivably  richer  in  mathetic 
content  than  can  be  any  outer  world  of  sense  such  as  the 


MATHEMATICS  307 

physical  universe  according  to  which  we  chance  to  have 
our  physical  being. 

This  mere  speck  of  a  physical  universe  in  which  the 
chemist,  the  physicist,  the  astronomer,  the  biologist, 
the  sociologist,  and  the  rest  of  nature  students,  find 
their  great  fields  and  their  deep  and  teeming  interests, 
may  be  a  realm  of  invariant  uniformities,  or  laws;  it 
may  be  a  mechanically  organic  aggregate,  connected 
into  an  ordered  whole  by  a  tissue  of  completely  defin- 
able functional  relationships;  and  it  may  not.  It  may 
be  that  the  universe  eternally  has  been  and  is  a  genuine 
cosmos;  it  may  be  that  the  external  sea  of  things  im- 
mersing us,  although  it  is  ever  changing  infinitely, 
changes  only  lawfully,  in  accordance  with  a  system 
of  immutable  rules  of  order  that  constitute  an  invariant 
at  once  underived  and  indestructible  and  securing 
everlasting  harmony  through  and  through;  and  it  may 
not  be  such.  The  student  of  nature  assumes,  he  rightly 
assumes,  that  it  is;  and,  moved  and  sustained  by  char- 
acteristic appropriate  curiosity,  he  endeavors  to  find 
in  the  outer  world  what  are  the  elements  and  what  the 
relationships  assumed  by  him  to  be  valid  there.  The 
mathematician  as  such  does  not  make  that  assumption 
and  does  not  seek  for  elements  and  relationships  in 
the  outer  world. 

Is  the  assumption  correct?  Undoubtedly  it  is  admis- 
sible, and  as  a  working  hypothesis  it  is  undoubtedly 
exceedingly  useful  or  even  indispensable  to  the  student 
of  external  nature;  but  is  it  true?  The  mathematician 
as  man  does  not  know  although  he  cares.  Man  as 
mathematician  neither  knows  nor  cares.  The  mathe- 
matician does  know,  however,  that,  if  the  assumption 
be  correct,  every  relationship  that  is  valid  in  nature 
is,  in  abstractu,  an  element  in  his  domain,  a  subject  for 


308  MATHEMATICS 

his  study.  He  knows,  too,  at  least  he  strongly  suspects, 
that,  if  the  assumption  be  not  correct,  his  domain 
remains  the  same  absolutely,  and  the  title  of  mathe- 
matics to  human  regard  "would  remain  unimpeached 
and  unimpaired"  were  the  universe  without  a  plan  or, 
having  a  plan,  if  it  "were  unrolled  like  a  map  at  our 
feet,  and  the  mind  of  man  qualified  to  take  in  the 
whole  scheme  of  creation  at  a  glance." 

The  two  realms,  of  mathematics,  of  natural  science, 
like  the  two  curiosities  and  the  two  attitudes,  the  mathe- 
matician's and  the  nature  student's,  are  fundamentally 
distinct  and  disparate.  To  think  logically  the  logically 
thinkable  —  that  is  the  mathematician's  aim.  To  as- 
sume that  nature  is  thus  thinkable,  an  embodied  rational 
logos,  and  to  discover  the  thought  supposed  incarnate 
there  —  these  are  at  once  the  principle  and  the  hope  of 
the  student  of  nature. 

Suppose  the  latter  student  is  right  and  that  the  outer 
universe  really  is  an  embodied  logos  of  reason,  does  it 
follow  that  all  the  logically  thinkable  is  incorporated 
in  it?  It  seems  not.  Indeed  there  appears  to  be  many 
a  rational  logos.  A  cosmos,  a  harmoniously  ordered 
universe,  one  that  through  and  through  is  self -com- 
patible, can  hardly  be  the  whole  of  reason  materialized 
and  objectified.  At  all  events  the  mathematician  has 
delight  in  the  conceptual  construction  and  in  the  con- 
templation of  divers  systems  that  are  inconsistent 
with  one  another  though  each  is  thoroughly  self -coherent. 
He  constructs  in  thought  a  summitless  hierarchy  of 
hyperspaces,  an  endless  series  of  ordered  worlds,  worlds 
that  are  possible  and  logically  actual.  And  he  is  con- 
tent not  to  know  if  any  of  them  be  otherwise  actual 
or  actualized.  There  is,  for  example,  a  Euclidean 
geometry  and  there  are  infinitely  many  kinds  of  non- 


MATHEMATICS  309 

Euclidean.  These  doctrines,  regarded  as  true  descrip- 
tions of  some  one  actual  space,  are  incompatible.  In 
our  universe,  to  be  specific,  if  it  be  as  Plato  thought 
and  natural  science  takes  for  granted,  a  geometrized 
or  geometrizable  affair,  then  one  of  these  geometries 
may  be,  none  of  them  may  be,  not  all  of  them  can  be, 
objectively  valid.  But  in  the  infinitely  vaster  world  of 
pure  thought,  in  the  world  of  mathesis,  all  of  them  are 
valid;  there  they  co-exist,  there  they  interlace  and  blend 
among  themselves  and  others  as  differing  strains  of 
a  hypercosmic  harmony. 

It  is  from  some  such  elevation,  not  the  misty  lowland 
of  the  sensuously  and  materially  Actual,  but  from 
a  mount  of  speculation  lawfully  rising  into  the  azure 
of  the  logically  Possible,  that  one  may  glimpse  the 
dawn  heralded  by  the  avowal  of  Leibnitz:  "J/a  mtta- 
physique  est  touie  mathtmaliquc"  Time  fails  me  to  deal 
fittingly  with  the  great  theme  herewith  suggested,  but 
I  cannot  quite  forbear  to  express  briefly  my  conviction 
that,  apart  from  its  service  to  kindred  interests  of 
thought  as  a  standard  of  clarity,  rigor  and  certitude, 
mathematics  is  and  will  be  found  to  be  an  inexhaust- 
ible quarry  of  material  —  of  ontologic  types,  of  ideas 
and  problems,  of  distinctions,  discriminants  and  hints, 
evidences,  analogies  and  intimations  —  all  for  the  ex- 
ploitation and  use  of  Philosophy,  Psychology,  and 
Theology.  The  allusion  is  not  to  such  celebrated  alli- 
ances of  philosophy  and  mathesis  as  flourished  in  the 
school  of  Pythagoras  and  in  the  gigantic  personalities 
of  Plato,  Descartes,  Spinoza,  and  Leibnitz,  nor  to  the 
more  technical  mat  he  matico- philosophical  researches 
and  speculations  of  our  own  time  by  such  as  C.  S. 
Peirce,  Russell,  Whitehead,  Peano,  G.  Cantor,  Couturat 
and  Poincare,  glorious  as  were  those  alliances  and 


310  MATHEMATICS 

important  as  these  researches  are.  The  reference  is 
rather  to  the  unappreciated  fact  that  the  measureless 
accumulated  wealth  of  the  realm  of  exact  thought  is 
at  once  a  marvelous  mine  of  subject  matter  and  a  rich 
and  ready  arsenal  for  those  great  human  concerns  of 
reflective  and  militant  thought  that  is  none  the  less 
important  because  it  is  not  exact. 

For  the  vindication  of  that  claim,  a  hint  or  two  must 
here  suffice.  The  modern  mathematical  concepts  of 
number,  time,  space,  order,  infinitude,  finitude,  group, 
manifold,  functionality,  and  innumerable  hosts  of  others, 
the  varied  processes  of  mathematics,  and  the  principles 
and  modes  of  its  growth  and  evolution,  all  of  these  or 
nearly  all  still  challenge  and  still  await  those  kinds  of 
analysis  that  are  proper  to  the  philosopher  and  the 
psychologist.  The  psychology  of  Euclidean,  non-Eu- 
clidean, and  hyperspaces,  the  question  of  the  intuita- 
bility  of  the  latter,  the  secret  of  their  having  become 
not  only  indispensable  in  various  branches  of  mathe- 
matics but  instrumentally  useful  in  other  fields  also, 
as  in  the  kinetic  theory  of  gases;  the  question,  for 
example,  why  it  is  that  while  thought  maintains  a 
straightforward  course  through  four-dimensional  space, 
imagination  travels  through  it  on  a  zigzag  path,  of 
two  logically  identical  configurations,  being  partially 
or  completely  blind  to  the  one,  yet  perfectly  beholding 
the  other;  the  evaluation  and  adjustment  of  the  con- 
tradictory claims  of  Poincare  and  his  school  on  the  one 
hand  and  of  Mach  and  his  disciples  on  the  other,  the 
former  contending  that  Modern  Analysis  is  a  "free 
creation  of  the  human  spirit"  guided  indeed  but  not 
constrained  by  experience  of  the  external  world,  being 
merely  kept  by  this  from  aimless  wandering  in  wayward 
paths;  while  the  latter  maintain  that  mathematical 


MATHEMATICS  31 1 

concepts,  however  tenuous  or  remote  or  recondite,  have 
been  literally  evolved  continuously  in  accordance  with 
the  needs  of  the  animal  organism  and  with  environ- 
mental conditions  out  of  the  veriest  elements  (feelings) 
of  physical  life,  and  accordingly  that  the  purest  offspring 
of  mathematical  thought  may  trace  a  legitimate  lineage 
back  and  down  to  the  lowliest  rudiments  of  physical 
and  physiological  experience:  —  these  problems  and 
such  as  these  are,  I  take  it,  problems  for  the  student 
of  mind  as  mind  and  for  the  student  of  psycho-physics. 
Regarding  the  relations  of  mathesis  to  the  former 
"queen  of  ail  the  sciences,"  I  have  on  this  occasion 
but  little  to  say.  I  do  not  believe  that  the  declined 
estate  of  Theology  is  destined  to  be  permanent.  The 
present  is  but  an  interregnum  in  her  reign  and  her 
fallen  days  will  have  an  end.  She  has  been  deposed 
mainly  because  she  has  not  seen  fit  to  avail  herself 
promptly  and  fully  of  the  dispensations  of  advancing 
knowledge.  The  aims,  however,  of  the  ancient  mistress 
are  as  high  as  ever,  and  when  she  shall  have  made  good 
her  present  lack  of  modern  education  and  learned  to 
extend  a  generous  and  eager  hospitality  to  modern 
light,  she  will  reascend,  and  will  occupy  with  dignity  as 
of  yore  an  exalted  place  in  the  ascending  scale  of  human 
interests  and  the  esteem  of  enlightened  men.  And 
mathematics,  by  the  character  of  her  inmost  being,  is 
especially  qualified,  I  believe,  to  assist  in  the  restoration. 
It  was  but  little  more  than  a  generation  ago  that  the 
mathematician,  philosopher  and  theologian,  Bernhard 
Bolzano,  dispelled  the  clouds  that  throughout  all  the 
foregone  centuries  had  enveloped  the  notion  of  Infini- 
tude in  darkness,  completely  sheared  the  great  term  of 
its  vagueness  without  shearing  it  of  its  strength,  and 
thus  rendered  it  forever  available  for  the  purposes  of 


312  MATHEMATICS 

logical  discourse.  Whereas,  too,  in  former  times  the 
Infinite  betrayed  its  presence  not  indeed  to  the  faculties 
of  Logic  but  only  to  the  spiritual  Imagination  and  Sensi- 
bility, mathematics  has  shown,  even  during  the  life  of 
the  elder  men  here  present,  —  and  the  achievement 
marks  an  epoch  in  the  history  of  man,  —  that  the 
structure  of  Transfinite  Being  is  open  to  exploration 
by  the  organon  of  Thought.  Again,  it  is  in  the  mathe- 
matical doctrine  of  Invariance,  the  realm  wherein  are 
sought  and  found  configurations  and  types  of  being 
that,  amid  the  swirl  and  stress  of  countless  hosts  of 
transformations,  remain  immutable,  and  the  spirit  dwells 
in  contemplation  of  the  serene  and  eternal  reign  of  the 
subtile  law  of  Form,  it  is  there  that  Theology  may  find, 
if  she  will,  the  clearest  conceptions,  the  noblest  symbols, 
the  most  inspiring  intimations,  the  most  illuminating 
illustrations,  and  the  surest  guarantees  of  the  object  of 
her  teaching  and  her  quest,  an  Eternal  Being,  unchan- 
ging in  the  midst  of  the  universal  flux. 

It  is  not,  however,  by  any  considerations  or  estimates 
of  utility  in  any  form  however  high  it  be  or  essential 
to  the  worldly  weal  of  'man;  it  is  not  by  evaluating 
mastery  of  the  processes  of  measurement  and  compu- 
tation, though  these  are  continuously  vital  everywhere 
to  the  conduct  of  practical  life;  nor  is  it  by  strengthen- 
ing the  arms  of  natural  science  and  speeding  her  con- 
quests in  a  thousand  ways  and  a  hundred  fields;  nor  yet 
by  extending  the  empire  of  the  human  intellect  over  the 
realms  of  number  and  space  and  establishing  the  do- 
minion of  thought  throughout  the  universe  of  logic;  it  is 
not  even  by  affording  argument  and  fact  and  light  to 
theology  and  so  contributing  to  the  advancement  of 
her  supreme  concerns;  —  it  is  not  by  any  of  these 
considerations  nor  by  all  of  them  that  Mathematics, 


MATHEMATICS  313 

were  she  called  upon  to  do  so,  would  rightly  seek  to 
vindicate  her  highest  claims  to  human  regard.  It 
requires  indeed  but  little  penetration  to  see  that  no 
science,  no  art,  no  doctrine,  no  human  activity  whatever, 
however  humble  or  high,  can  ultimately  succeed  in 
justifying  itself  in  terms  of  measurable  fruits  and  ema- 
nant  effects,  for  these  remain  always  to  be  themselves 
appraised,  and  the  process  of  such  attempted  vindica- 
tion is  plainly  fated  to  issue  only  in  regression  without 
an  end.  Such  Baconian  apologetic,  when  offered  as 
final,  quite  mistakes  the  finest  mood  of  the  scientific 
spirit  and  is  beneath  the  level  of  academic  faith.  Sci- 
ence does  not  seek  emancipation  in  order  to  become  a 
drudge,  she  consents  to  serve  indeed  but  her  service 
aims  at  freedom  as  an  end. 

Man  has  been  so  long  a  slave  of  circumstance  and 
need,  he  has  been  so  long  constrained  to  seek  license 
for  his  summit  faculties,  in  lower  courts  without  appeal, 
that  a  sudden  transitory  moment  of  release  sets  him 
trembling  with  distrust  and  fear,  an  occasional  imperfect 
vision  of  the  instant  dignity  of  his  spiritual  enterprises 
is  at  once  obscured  by  doubt,  and  he  straightway 
descends  into  the  market  places  of  the  world  to  excuse 
or  to  justify  his  illumination,  pleading  some  mere  utility 
against  the  ignoring  or  the  condemnation  of  an  insight 
or  an  inspiration  whose  worth  is  nevertheless  immediate 
and  no  more  needs  and  no  more  admits  of  utilitarian 
justification  than  the  breaking  of  morning  light  on 
mountain  peaks  or  the  bounding  of  lambs  in  a  meadow. 

The  solemn  cant  of  Science  in  our  day  and  her 
sombre  visage  are  but  the  lingering  tone  and  shade  of 
the  prisonhouse,  and  they  will  pass  away.  Science  is 
destined  to  appear  as  the  child  and  the  parent  of  freedom 
blessing  the  earth  without  design.  Not  in  the  ground 


314  MATHEMATICS 

of  need,  not  in  bent  and  painful  toil,  but  in  the  deep- 
centred  play-instinct  of  the  world,  in  the  joyous  mood 
of  the  eternal  Being,  which  is  always  young,  Science 
has  her  origin  and  root;  and  her  spirit,  which  is  the 
spirit  of  genius  in  moments  of  elevation,  is  but  a  sub- 
limated form  of  play,  the  austere  and  lofty  analogue 
of  the  kitten  playing  with  the  entangled  skein  or  of  the 
eaglet  sporting  with  the  mountain  winds. 


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